Solution of the Problem of Homing in a Vacuum with a Point Mass EXTERNAL MEMORANDUM NO. 19 Project MX-794 (AAF Contract W33-038 ac-14222) Prepared by F. D. Faulkner Approved by R. V. Churchill Supervisor, Mathematics Group February 1, 1949

AERONAUTICAL RE-SEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 19 This report was prepared with the active participation of the following members of the Mathematics Group: C. M. Fowler W. A. Wilson Page ii

ENGINEERING LIBRARY AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 19 CONTENTS Section Page I INTRODUCTION AND SUMMA'RY ^ 1 II FORMIULA.S AND EQUATIONS 9 III THE NMIETHOD OF SOLUTI0ONJ OF T.HE HOMING PROBLEM 15 IV DISCUSSION OF POSSIBLE TIM:,ES OF INTERCEPTION A.ND OF THE PATHS OF MINIMLUM FUEL CONSUDPTION l. 19 V EXAlMPLE OF A HOMING PROBLEM 22 VI'THE SOLUTION OF A HOMTING PROBLEOM IN SPACE 25 VII BASIC TIHE-ORY UNDERLYING TTHE METHOD OF SOLVING THE HOMING PROBLEM 31 VIII DISCUSSION AND LIMITA.TIONS OF THE NME T-TOD 3 5 IX SO0':"E ~A, PLICATIONS OF TlHE i',/'THtOD TO PROBLEMS OF DESIGN 38 X TWO THEOREMS.ONT HOMING 43 -' Page iii

AERONAUTICAL RESEARCH'CENTER,- UNIVERSITY -OF MICHIGAN Report No UMM 19 ILLUSTRATIONS Figure facing Page I Curve for Distp tace versus Time in Dimensionless Form....... 10 II Coordinate Set and Target Path......... 14 III Use of Curve in Solving e Problem...... 16 IV Use of Curve in Solving a Problem... 17 V Some Possible Intersections of Curves for s-Xand S*-..................... 18 VI Target Position, Indicating Possible Homing'Points.... 22 VII Solution of Homing Problem....... 23 VIII Solution of Homing Problem........... 28 IX Grids for Various Values of ro and Fixed Target C onditions........ 38 X Determination of ro for Interception and of Lower Bound to Fuel Consumption........ 39 Page iv

:AEJRO NAUTICAL R'ESREARCH C-ENTER. ~ UNIVERSITY OF MICHIGAN Report No UMM 19 I. INTRODUCTION AND SUMMARY It is the main purpose of this report to give a simple, general solution to the problem of homing with a point mass in a vacuum. This report is associated closely with Report No. UMM 18 on the subject of homing with minimum fuel consumption. The background of the problem is given there in some detail and we shall not go into it here. By homing is meant the application of thrustI to the first particle, essentially a rocket, called the craft, in such a way that its position will at a later time coin'cide with that of the second particle, the target. The target is assumed to be in free flight or, in the later sections, following any prescribed course. The method of this report also gives the method for. determining the paths of minimum fuel consumption and their existence. It has important applications for the designer in 1The thrust is a force gained by the emission'of part of the mass of the craft. The average velocity of these particles over any time interval is assumed to be constant, designated. by c, the effective gas velocity. Page 1

ASERON'AUTIC'AL R ESE:ARC;H CE NTER — UNIVERSITY OF MICHIGAN Report No UMM 19 determining craft specifications for homing. The method also gives the engineer an approximate answer to many other problems that arise in rocket work. Some of these approximations are given and the sources of errors due to the approximations are pointed out. A. Points Particularly Important to the Homing Problem The method can be used whenever the thrust is a specified function of time and when the target course can be predicted. This includes the case in which the acceleration due to thrust is a specified function of time. The key to the method is this: a single grid can be drawn up which, except for parameters of the particular rocket chosen, completely describes the motion of all rockets which have a similar thrust. As tan example, all rockets whose thrust is constant have similar thrust. Because of its importance the work is carried through for this case -with supplementary explanations of the direct generalizations. The grid is drawn up in dimensionless form. The initial conditions, that is, the initial relative velocity and position of the target with respect to the craft, are expressed in terms of the parameters of the rocket. This determines a curve which is superimposed on the grid. 1Explained in detail in the text. Page 2 -

AERON.AUTICAL RESEARCH'; CENTER,UNIVERSITY OF MICHIGAN Report No UMM 19 The following quantities can be read directly from the graphs.; 1. The times when homing can be effected. 2. The burning time (the duration of thrust) and fuel consumption corresponding to any chosen homing time. 3. The existence (or non-existence) of a minimum path1. 4. The homing time, burning time, and fuel consumption corresponding to the minimum path. The me-thod appears to be well adapted to use in the field. Thile important reason is this: the characteristics of the craft do not need to be specified beforehand; they need only to be specified at the instant h'oming starts. This seems important if we consider that homing will take place in a succession of steps a's the information becomes more and more accurate.:Each step will begin with an amount of fuel which cannot be predicted beforehand but which can be computed, since the initial weight (at the beginning of the first step) and the amount of fuel' burned in each -step will be known as soon as the step is over; lLet us refer to paths of minimum fuel consumption as minimum paths.

AERONAUTI GAL'RE S EARCH CE.NTER - UNIVERS ITY'-Ct MICHIGAN Report No UMM 19 As shown in UMIJ-18 the problem of homing is entirely a problem in relative motion. In the early stages of the homing problem it may be necessary to use information from earthbased equipment. In this case there are at least fifteen parameters in the problem: the initial position of the craft (three coordinates), its initial velocity (three more parameters), the position and velocity of the target, the thrust of the craft, the effective velocity of the jet, and the initial weight of the craft. The second important feature is this. It is shown that the fifteen parameters above can be combined to?yield three significant parameters; these three parameters completely specify the problem. 2 This simplifies the over-all problem greatly. Of course any solution' in- terms of initial values must be reduced to this form eventually, either explicitly or implicitly in that the computations' carried out are entirely equivalent. Immediate, reduction to this form removes the mysticism from the solution. 1These can'be expressed in numerous other equivalent ways. 2There is a fourth parameter (a sixteenth parameter in the first set) which corresponds to the maximum allowable burning time, or to the total amount of fuel which the rocket has. Page 4'

AEERONAUTICAL RESEA.RCH- CENTER - UNIVERSITY OF MICHIGAN Report No UMMI 19 B. Points Particularly Important in the Design of a Homing Craft The method can be used to find the following information of particular interest to a person designing a homing craft; For given target conditions and known jet velocity one can determine the following: 1. The times when homing can be effected by burning all the way.l 2. For a chosen homing time, the fuel consumption corresponding to burning all the way (if it is possible). By virtue of properties (1) and (2) we can draw a graph which will show the lowest burning rate whi.ch can effect homing (and the homing time corresponding to this burning rate). There may not be a lowest burning rate; the burning rate may decrease steadily as homing time increase's. 3. For a chosen homing time, the fuel consumption corresponding to thrust applied as an impulse. We will see that if the average velocity during homing exceeds c, the jet velocity, then homing cannot be effected by burning all the way at a constant rate. Page 5

..A1ER.ONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN.Report No UMM 19 Familiarity with the method will reveal other important properties to those who use it. liany of the properties of minimum paths can be shown directly from the graphs. In UMM-18 certain principles of minimizing fuel consumption were given. Thiese could equally well have been called principles for maximum perfo-mrance; that is, how to achieve maximum performance with a given craft and a given amount of fuel. The two most important pri ciples were that 1. thrust must be fixed in direction (the first fundamental principle for minimizin[g fuel consumption), and 2. thrust rust be high during the early moments of the homing, then it must be cut off (the second fundamental principle for minimizing fuel consumption). The homing paths determined here are found under the assumption that these principles are observed. Fortunately these principles, articularly the second, seem to be quite natural ones to follow when the initial velocity and initial position are known. For a detailed analysis of a particular craft it is often simpler not to change to the dimensionless form, that is, to leave distances expressed in feet, times in seconds, etc. Page 6

AERONAUTICAL RESEARCH CENTER UNIVERSITY OF MICHIGAN Report No UMiM 19 C. Obvious Generalizations and-Limitations of the Method The method can be used any time that the vector of relative position can be broken down into two components, one independent of the thrust and the other due entirely to thrust., Simple extensions of the methods are given to solve the following problems. 1. The problem wihere the target follows a known dodging course. 2. The problem of sending a rocket to a fixed point in -space. One can read directly from the graph all the quantities mentioned previously such as the time when the craft can home (or arrive), fuel consumption, the fuel consumption corresponding to burning all the way and corresponding to an impulse, the determination of paths of minimum fuel consumption, if they exist, etc. The engineer and the designer are frequently called upon to make quick estimates and approximations. One aim of this report is to provide them with a method for getting these. Therefore we point out the limitations and the places where approximations are made, so that they can evaluate their results. In the first place we treat the point mass problem. Thlis is a good approximation so long as the times required to| rotate the craft can be neglected and the displacements resulting Page 7

AERONAUTICAL -RESEARCH CENTER ~ UNiVERSITY' OF -MICHIGAN Report No UMVIM 19 from the rotations can be neglected in comparison to the other times and displacements. In the second place it is assumed that the action takes place in a vacuum. This approximation is good as long as the acceleration due to interaction with the air can be neglected. In many cases the method gives...a —first approximation, Using this, we can compute the aerodynamic forces and superimpose them as a * perturbation upon the first approximation to yield two resultss, one above the true value for distance and velocity and the second below the t:rue value. |Part of the performance analysis in Project WIZARD Phase One Report. was carried out in a similar way. Some detailed computation should be carried out to give a measure of the valid- l ity of the answers. The third approximation is this. It is assumed that the target and the craft are close enough together that the acceleration due to gravity.is the same on each. This introduces an error in the'acceleration of approximately one foot per sec.,.per sec. for each three hundred thousand feet distance between them when they are near the earthts surface. When we have accelerations from thrust of the order of three to ten times the acceleration of gravity, this seems negligible. For a long period of free flight this may introduce an appreciable error. Page 8

AERONAUTIECAL RESEARCH CETER- UNIVERSITY- OF MICHIGAN Report No UMM 19 II. FORMULAS AND EQUATIONS A. General Formulas The equation of motion of a rocket in linear motion in a field-free space can be written (2. 1) Pcr w where w is the velocity, c ia the effective velocity of the emitted gas, usually assumed constant, r: is the: burnt fuel ratio = m is the mass of the fuel consumed at any time, Mo is the initial mass of the rocket complete with fuels, M is the mass: at any tim'e (Mo = M + m) Let us use dots over variables to indicate the derivative with respect to the time t. the thrus.t force is given (in magnitude) by (2s2) T = mc = - Mc. Page 9

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UI.VIM 19'fVe can inte-grate equation (2.1) to get the well-known forrmula for velocity, (2.3) w = - c n (1- r) if the rocket starts at rest. If thrust is constant we can irnte:!rate equation (2.3) to.et distance (2..4) s(t) = ct + (1 - 1) ln (1 - r)] r during burning., if the craft starts at rest. WTe define -the burning time tl by the relation r= r a constant, for t < tl (2,5) r = 0 for t > tl. We have the second formula for distance (2.6) s(t,t1) = - ct in (1 - r1) + ct1 [i + in (1 - ri)] rC for t > t1. This eauation includes equation (2.4) if we define tz = t duringfn burn.ing. B.l, Dim e n ensio nles s Form o-f t he eqiu t'ions of -Mot iol Let us defi'n-,e:the dimensionliess quentities t,- = -rot (2. 7 ) w-:'- -= w = s....... Pa;ge 10

> 1.0 0.9 o i ~~~~~~~0.81 1 I I 1 I I I 0 0.7. Ci. Thrust Constant for t*< t* Thrust Zero for tj> t*1~a 0.9 o~~s I I r I I. I I I I ~~~~~~~~ I/ n I I ~ ~' I C) 0.8 Thrust Zero for t > t 0.5,z 0.4 0.2 w~~~~~~~~~~~~~~~~~~ r~ d. 0.1 _ _ _ _ _ _ _ 0 0.1 0.2 0.3 0,4 0.5 0;6 0.7 0,8. 0.9 1.0 - Fig. I Curve for Distance vs. Time in Dimensionless Form.

AERONAUTICAL RE-SEARCH CENTER UNIVERSITY OF MICHIGAN Report No U1I 19 the eaquations ab'ove become (2.8) w:- - in ( - t), (2.9) s-(t) = -t:- + (1 - t:-) In (A1 - t:-), and (2.10) s*:-(t:-,t1*-) = - t- in (1 - t') + in (1 - t- ) Note that these formulas, which are the same for all rockets whose thrust is constant, completely describe the motion of the craft except for the parsameters c and ro.o Figure I is a graph of s*(t —,t1-:) versus t*e with t e* as a parameter. We see the following interestinS property: if it were possible to burn the craft up with the last particles delivering the same momentum, the craft would, in its last moments, reach an infinite velocity, but it would travel only a finite distance. It has a second interesting and useful prol-erty. By direct substitution we &et the relation (2.1 1) s- (l,tle ) = t This is used in the following way.'Ve shall want to find theI point of tangency to the curve for s —(t* ). By noting the intercept of the tangent with the line t:: = 1 we get the value of t*-: at the point of tansency much more accurately than we could by examining the intersection of the tansent Writh the curve. Page 11

'AERONAUTICAL REStEARCH' CENTER -UNIVERSITY OF -MICHIGAN Report No UMMI 19 Intuitively we define 1/ro as the burnup t ime, that is, the time it would ta.ke the craft to burn up entirely if it cont i'nued at the initisl rate. We define c/Jo as the burnup distance, the distence the craft would travelt if the same equation of motion (2.1) held until the craft burned up. The dimensionless terms above then express times in terms of the burn-up time, velocities in. terms of the jet velocity, and distances in terms of the burnup distance. If thrust is constant then r = t — during burning and C. Similarity Considerations The previous work is a simple application of similarity coisiderations. These are frequently used to reduce many particular problems to a small numb-r of general problemsl. Let us cons-',der the motion of two craft. Let us denote quantities associated with the second craft by Greek letters. Let t'ie burned fuel ratios be r and. resgectively. Definition. If the unit fuel, consumptions r and psatisfy the relation (2.12) p(t) = r(kt); 1These are particularly important in problems in compressible flow. See, for example, Dodge and Thompson, Fluid Mechanics, New York: McGraw Hill, 1937, pp 420 ff. Page 12'

AERONAUTICAL RE SEARCH CCENTER; UNIVERSITY OF MICHIGAN Report No UNMt 19 then the thrust of the two rockets is similar; the two rockets a re said to burn in a simila r manner.. The curves for r and e vs. t are similar in the usual sense of the definition of similar curves in dynamics. For linear motion we see that the velocities w and. w satisfy the relation (2.13) (t) - n (1 -(t)) |= - 7 ln (1- r(kt)) - w(kt) where c and' are the respective effective g.s velocities. If c then (2.13 ) w (t) = w(kt) If we differentiate in equation (2.12) we see that (2.14). (t) = k r(kt, and that the accelerations a and a. sactisfy the relation (2.15) Q(t) (2.15) ) 1 - p(t) 1 - r(kt) k -a (kt) PaCe 13

AERONAUTICAL RESEARCH:CENTER - UNIVERSITY OF MICHIGAN Report No UMM 19 If c = I, then (2.15'). o(t) = k a(kt). If we integrate in equation (2.13) we find that (2.16)' - (kt) where s and o- are the distances for the two craft. ThLe craft. are assumed to start from rest et the origin. If = c, then (2.161) -(t) = s(kt) k It is clear that the curves for acceleration, velocity,.distance and fuel consumption as functions of time for all rockets that have similar thrust differ only in the scale used along the axes and that points on one of these. curves for any rocket of the set can be obtained from the corresponding curve for any other rocket of the set by the proper use of the factors k and v/c, Hence a single curve can be drawn up in dimensionless form obtained by dividing t by some parameter corresponding to 1/r of paraEgraph B and dividing s by the corresponding quantity involving c. A curve of the dimensionless quantities st as a function of t_* with parameter _t-, is obtained corresponding to the curves of Figure I. In general a second curve is needed to convert | to r. ------ -—' Pag.e 14

0 Point of Interception Target at Target Path Time t U Target:|.(t=O). 00 m t!1 01 Craft (t=0) ] Fig:r Coordinate Set and Target Path 0-. (3

AERON:AUTIL RESiE"ARCH CENTER - UNWVE;RSITY OF MICHIGAN,....... Report No UMM 19 III. THE METHOD OF SOLUTIO N OF TTE HO.ING PROBLEM.We shall choose our coordinate set so that the craft is |originally at the origin wJith zero velocity' and its motion is due entirely to the thrust applied.'We choose thle coordinate set so that the target is initially st (Xo,YQ)with its velocity (U,O) parallel to the x-axis as in Figure II. This can always be donel. The co.ordinates of t.he target position in this system are X = Xo + Ut - U(t - t'), (3.1) Y =Yo,,. Xo where t' = - ~; t' is the time when the target i.s ne.arest to the orig-in. In polar form the target position becomes S ~/X2 y2 (302)'Yo (3.2) arctan.o + Ut 1See UMM-18, the appendix. Page 15

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UIM 19 Wle can write the first of these es (3 3): S2 - U2(t tt)2 y 2 the equation of a hyperbole in the tS^plane. The creft position can be expressed in polar forr () and we note th1t s is given by the forrmulas of Section II. TWe can change s to its dimensionless form s* end then the grid of Figure I is the graph of s*(t*,tl-). Only a limited amount of fuel will be aveilable and this will plce a limit on t, end t. Thls deterrmines the hecvy tanennt curve of Fiure III, celled the curve of maximuma performance, which se cannot exceed. In Figure III this maiXnum value of 2 wes taken arbitrarily as.5 corresponding to e Suel vweight of one hPlf the initiel gross weight. Nor let us charnge the quentities S X, Y, U send t to a dimensionless form in the s ee mnner n t he preceding section, by multiplying-r t imes by ro, dividin velocities by c and distances by c/ro. lWe denote the corresp nding dimensionless terms by * as before. Equation (3. 3) becormes;7: (v.4) U - Y 2 In the t*S* —plan'e this is the equation of a hyperbole with center at (t?'*,O) with asymptotes of slope + U*i- and semi-transverse'Pa —-~e. 16

L ~ ~ ~~~~- - 0 - -~- - - - - C 1.6 j 0.21.0 1. 1..6.8 2.. 2. 0 - I I I I I I I ) 1 ( -1A~~ m - -, -'IO 1-L O/~~~~~~~~~~~~~~~C ~~~~~of ~ ~ ~ ~ i~. s fCrei oligaPolme I~~~~~~~~~~~~~~~~~~~~~~~l 0.4 t i - tjrf.001 0.2 I I I0 2 0.811 1.:0.2 1 0, 4 ~~0.6;, 1.9 1.4 1.6 1.8 2- 2~2 1 2. 4 2.6 = ~~~~~~~~Max-t -Min t2 - ax t2Fig-M Use of Curve in Solvring a Problem ~

1.6 C) 0.4__ _ _' 0.2_ ___ __ l * * 0.8 _______' ____ t.8, V - - 0.6 22____: /A 0.4~~~~~~~~~~. Coresoni 0'2 ____ ____ —f~- --— C~ —--- t, Corresponding:07 Gorsndn to t*~~~O7 ~to a Min.Path Fig. N. Use of Curve in Solving a Problem

"AERONAUTICAL RESE-ARCH CENTER UNIVERSITY OF MICHIGAN Report No UMM 19 axis o Let us draw the graph of equation (3.4) on the sarnme set of coordinates as those for equations (2.9) and (2.10) (see Figure III). Since S*- is essentially a distance we shall consider only the upper branch of the curve. Jiow all points of the S. —curve which lie below the curve of maximum performance represent possible homin. times and homing points. For a chosen t:- in this range we find t -as follows (see Figure IV where we chose t arbitrarily as 0.7). Draw the tangent to the si* —curve from the point on the S-3-curve. The point of tangency determines ti":. As remarked in Section II, it is difficult to determine the exact point of tangency directly but we can use the relat.ion l(2. 113 - " - s(,t-) = t',to find t1- quite accurately. For t2- =.7 we see from Figure III that t-:- =.335. We see thate tlere is a path of minimum fuel consumption. The corresponding values of t2- and t- are.93 and.225 respectively. We see that for this example homing is possible for.65 < te* < 2.25. For the extreme values, tas- =.5, and it is less for the points in between..... Pa.ge 17

AERONAUTICAL.. RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 19 We have satisfied the first condition of homing, that' s = S. The second condition is (3.7) (t).Yo Y' o arctan ( arcan + u Xo + Ut2 XO-"' U-:'Lt2s:If it is sim.-pler to apply thrust at some particular angle than at any other, this condition can be- satisfied, the corresponding homin, tirmes be de'termined (from equation 3.7) and the burning time be determined from the graph as in Figure IV. If we choose our axes ~differently, equation (3.3) will have the form S2 - W2(t - t')2 = S2min where W is the magnitude' of the initial. relative velocity, Smin is the distance from the origin to the target when the target is nearest, and t' is the time correspondin., to S = Smin. For our choice of axes we see that U W= and Yo = Stin' - Page 18

AERONAUTICAL RE-SEiARCH:-CENTER -' UNIVERSITY OF MICHIGAN Max. Performance Curve / t Min. tt t Fig Ma l l Fi% Zb: XS fig. Zc l l FjiJdl Fq in o /sil nefeto o uv o Sod Fig. Z Some Possible Intersections of Curves for S* and s

At'RONAUTICAL''RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 1-9 IV. DISCUSSION OF.POSSIBLE TIMES OF INTERCEPTION' AND OF THE PATHS OF MINIMUM FUE-L-C ONSUMPTION. The curve of Figure I is the same for all rockets which have similar thrust, except for the maximum fuel consurmption (or burning time). The conditions of homing then depend only upon the hyperbola. We have just seen that we could: determine at a glanrce whether.homin'g, could be effected or not: honing was possible at all times when the curve of maximum performance was above or touching the hyperbola. It is equally easy to determine whether.. or not a path of minimum fuel consumption exists: the necessasry and sufficient condition is that the curve of si(t*:-) and the,hyperbols have a cormmnon tangent such that t'i: < t2 —. The burning time t_ corresponds to the point of tangency to the curve of s*(t:-:), and in this case the homing time t2 corresponds to the point of tangency to the hyperbola (see Figure V.e). For example, if the asymptote of pQsitive. slope does not cut the curve of s*:-(t —), then there:is a prth of minimum fuel Page'19

AERONAUTICAL RESEARCGH' CENTER - UNIVERSITY OF MICHIGAN Report No UMNIi 19 consumption. This condition is not necessary for s path of minimvurm fuel consumption, as is shown in Figure V.b. There ray also be isolated points of interception such that t,- = t2* and such that the two curves have a common tanogent as in Figure V.c. These are not paths of minimum fuel consumption since homing is not possible-for any neighboring points (times). In this case the two curves for s-* and S-* may or may not have a common tangent later, corresponding to a relative minimum. If they have', the relative minimum requires more fuel than the path for the isolated. homing time. It can also happen that the curve for S3- may cross the curve for s-' (during burning) three times and that the asymptote does not cut the s*-curve. In this case there are two relotive minima, the lower one being elso an absolute minimum to fuel consumption. These are the cases referred to in UMM-18, p 12. If there is no common tangent to the two curves, then there is no path of minimum fuel consumption. There is a critical fuel consiumption r" = 1 - e which can be approached erbitrarily close. In this case fuel consumption goes up as homing time decreases an-d, vice versa, as homing time blecomes infinite the unit fuel con sumption r approaches r". On: the graph this is represented by the curves of s* and of Pagse 20

: AERONAUTI CA:L -E S EARCH CENTER - UNIVERS ITY HsOF MICHIG'AN Repoart No UMM 19Si*:- being a symptotically- parallel. That is, the curve for s':- is parallel to and lies on or under the as-ymptote of the curve for It is not difficult to show t the pths found bove It is not dlfficul tto slzow th^t the pstths found Jsbove are pcths of minimum fuel consumption, for the curves-describe all paths for whiich thlrust is-fixed in direction rnd for which thrust is hizh at first, then zerol. Herce we need only consider thlese paths and:find the one for; w:':ich r -t1- is lowest. InI t;he case where it is not possible to th:irottle the burningl, the curve of maximum performance is the onI:r curve allowed. This is the case for the solid fuel rockets aveilable t o d a T.. Several othler properties co'ie of considerin, hyperbolas. For example, as pointed out eabove, thlere may be no intersection of the two curves or tlhere msy be as rmsany as th-ree, e11 represent"in, possible'o "lomin, s i-tuat ions. 1Tllhese principles were eL)ressed and proved in U]/IM-18 to be necessary if fuel consurilption was to be minimized. Pase.-21 - l.

AERONAUTICAL RESE'ARCH CENTER - UNIVERSITY'OF MICHIGAN Report No UMM 19 V. EXAMPLE OF A HOING PROBLEM As a specific example, cornsider a rocket with an initial weight of 1500 pounds, thrust 15,000 pounds, specific im:pulse 250,' (and -dW/dt = 60 pounds/sec). Then c = Ig = 8000 (ft/sec) -dW 60 dt/O 1500 IC....8000' c 8000- = 200,000 (ft) r.04 Consider an incoming tar-et initisally 100,000 feet distant, relative -velocity 10,000 ft/sec, the -velocity vector 109 (1700) from the position vector. Change to dimensionless form. Then the initial range becomes, in the dimensionless form, - 100, o00 200,-000 velo'city becomes U — = 10,000 = 1.25; 8,000 Page 22 - *

Y~~~~~~~~~~~~~~~~ Target Path.. Target Position Target Position Initially P5 N.4.3 for Z.2 t. (X,'Z N Min. N M 100 0.2.3 4.5 - ~ 1! Fig. 9 Target Position, Indicating Possible Homing Points 1O 0

1.0 0 0.9 ~~~~~~~~~~~~~~~~~~~0,8 0.8 - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~j'1 0.7 0.6 * C0 0.5 _ _ _ _C " 0.4 " / t ~~0.4,~NOTE:~.'Min. Corresponds to / Minimum Fuel Consumption 0.3 _ _ _ _ _ _ _ _ _ _ 0.2 I-4 0~~~~, VI.. I,=,1 0.1 ___ _ 01~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~7 0 0.1 0.2 03* 0.4, 0.5 0.6 0.7 0. 09 1.0. Min. t, Min. t. t* Fig.. Solution of Homing Problem zg

AERONAUTICAIL RESEAR CH - CENTER - UNIVERSITY OF MICHIGAN Report No UMM 19 and t tX = 25 Now one way to solve the problem would be to obtain the equation of the hyperbola and plot it. However, s simpler way is to take the grrph of the treget position versus "time" in the dimensionless form, Figure VI ani-d lay off the target position as a function of t -, say in intervals of.1 for t* —. Then the distance from the origin to the point is the ordinate of the hyperbola as a function -of "time". Transfer these points onto the curve of rocket performance with a pair of dividers, Figure VII..This is a ruler and compass construction of the hyperbola' thet can be done quickly. The asymptotes are also known from the time when the target is nearest to the origin and its velocity. Figure VII shows the hyperbola sketched in. As many points as are wanted may be found. In this particular case homing could take place for.39 - t:2 -.5, and a burning time.3 v tl- <,5. Since t corresponds to r, the minimum fuel consumption would be 450 pounds to cause interception. The corresponding angles may be found from Figure VI as Gmin = 860 ~ 9 - 1470 = 9max since homing time would be known. The method may be made as accurate as desired. It seems probable that the parameters could all be fed into a machine that would construct the hyperbola at once PagWe 23....

AERO AUTICAL S EARCH ENTER - UNIVE R ITY OF MICHIIGAN Report No'UMMI19 and solve the equation. Either this machine would have to be in the craft (this seems unlikely) or the information would have to be converted to information for the craft in space.. Thet is, a ground-based comr,)uter would have to obtain -position and velocity from some source, and convert this to rngle and burning time.. Page 24

-AERONAUTTICAL. RESEARCH CENTER -I UNIVERSITY OF" MICHIGAN Report No UMM 19 VI. TIE SOLUTION OF A HOMlING PROBLEM IN SPACE The solution of the homing problem in Section V was given for the problem after it had been reduced t-o the two-dimensional form.1 For pract- ical purposes, the solution can be carried out alr:ost as simply for the three-dimensional form. Let us work out a second example to show the rethod. The computations involved are simple. The hyperbola is drrwn, a homing time is selected froml the grsph, the burning time is then read fromr the graph, and then three eauFtions give the direction cosines for thle thrust. Consider the problem in the three-dimensional form. The relative position of the terg.::t is given by X = Xo + Ut (6.1) Y = YO + Vt Z = Zo +' t in our coordinate system. Pa,;_e 25

AEIRONAUT-ICAL. RESEARCH CENTER - UNIVERSITY OF'MICHIGAN Report No UMM 19 Its distsnce from the origin is (6.2) S + Y I= t(Xo )2 + (y + Vt)2 + (zo + wt)2 We shall consider only thrust fixed in direction. Hence the position of the craft is given by x - s(6.3) y =9 z Z = V S where \, /, v are the direction cosines of the thrust, and s = adt2 For the cese of constant thrust applied for a time tl, with a specific fuel consumption r, s is given by (2.6) s(t,tl) = -ct ln(l - r) + ctl + ln (1- r for t r= tl The condition of homing is that there be a time t- t l tl such that Page 26

AERONAUTICAL-, RESEAR'CH CENTER - UNIVERSITY:fOF MICHIGAN Report No UMM 19 s(t2,t1) = S(t2), and (6.4) Xo + Ut2 YO + Vt2 ZO + rWt2 Consider now a specific problem. Let the ori;-.in be on the earth with the z-axis vertical initially, and the x- and yaxes any direction to form an orthogonal set. Let the axes be fixed in orientation like the axis of a free gyroscope. Let the craft be originally et the point xo, yQ, z0 with components 400,000, 100,000, 400,000, (feet), with velocity components, Uo, Vo, Wo 3,000, -1,000, 2,000 (ft/sec). Let the target be initially at the point X, Y', Z', with components 464,000, 160,000, 448,000, (feet), and with its velocity components Uo, Vo, ao equal respectively to -3,000, -7,200, -3,056, (ft/sec). Then the relative position has components Xo, Y, Z, 64,000, 60,000, 48,000, (reet), Page: 27

AERONAUTICAL: RESE ARCH CENTER - UNIVERSITY OF,MICHIGAN Report No-UMM 19 and the relative velocity has components'U, V, WV, -6,o000, -6,200, -5,056, (ft /sec). (These perticular numbers were chosen to make the distance between the two craft 10O,'000 feet and the relative velocityi 1,000 ft/sec. ) Let the craft weigh 1500 pounds initi lly, let it develop| 15,000 pounds thrust, let it carry 400 pounds of fuel, let the specific impulse I be 200. Then the effective gas velocity is c = 6,400 (ft/sec). Its burning rte- r is, 15,000. = 1500 I:' 0 " and the rimaximurm burned fuel ratio is 400 rma. - -.2666 rmax - 1500 The maximum burning time is t 2666 5.33 sec. tlmax -.05 Let us reduce to dimensionless form by dividing distances by D/ /o = 128,000 and times by l/ro (= 20). Pafje 28 --

1.0 F 0.9. 0.8 0.7 ~~~~~~~~~~~~~~~~~~~~~~ o.s9 _ _ _ _ A~~~~~~~~~~~~~~~~~~~ 0.8__ _\ 0.7 4 _ _ 0.3, C)l~ ~~b 1 I I I I I I~~~~~~~~~~~~~~~ I. I I I C~~~~~~~'1~f lot~~~~~~~~~~~~~~~~i,Ce~~~~~~~~~~~~~~C 0.5 2' t 0.4 0.2 _ _ _ _ _ _ _ _ _ _ _ _ _ _ - I a I I I.. H 01 t0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1O. t= it Fig. TT Solution of Homing Problem

AERONAUTICAL RESEARCH CENTER UNIVERSITY OF MICHIGAN Report No UBIl 19 The curve of rocket performrnce is the graph of Figure I. Wve plot - V\/(XO + Ut)2 + (YO + Vt)2 + (zo +wt)2 128, 000 as a function of t* (= rot) on the same graph (Figure VIII), We see from the graph that homing is possible for.445' —.585, that is for 8.9 t 11.7 (sec). The minimur fuel consumption is given by t-= r =.085 with a homing time of t = 10 (sec). The direction cosines.of thrust, corresponding to these homing times are given by equation (11.4) / t2 V 8.9.882.401.249 10.776 -.388 -.497 11.7 -.348 -.704: -.626 Page 29

AERONAUTICA:L RESEARCH CENTER -'UNIVERSITY OF, MICHIGAN Report No UMM 19 The point of homing may be found by the target position at the time of horfing. If the homing time chosen is 10 sec., the |point of homing will be IX = Xo' + Uot2 = 434,000 Y = Yo' + Vot2 = 88,000 Z = Z0o + Vot2 - 2 gt22 = 415,890 2 The gravitational acceleration is 30.81 +.10 ft/sec.2 on both target and craft1. The error by assuming it to be the same on each is less than 10 feet displacement during homing. These distances wou(.ld be measured in a non-rotating reference frame to avoid the complications of centrifugal forces and Coriolis forces. The ang:ular error in the above problem (the supplement of the angle between the vectors of relative position and relative velocity) can be found iby the methods of analytic geometry, using the formula cos p =-; (XAk2 +/lz2 + VI 21 where 1,, V1, are the direction cosines of the first vector, etc. The sajlle, for this problem turns out to be 2.95~. -lTiese values are based on the value 32.16 for the gravitational constant at sea level. - Page 30

AER ONAAUTICAL? RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 19 VII. BASIC THIEORY UNDERLYING THE METH-OD OF SOLVING THE HOMING PROBLEM In this section the essentials of this method of solving the homing problem will be pointed out. In general we will have a rocket craft which we wish to move from one point to another by means of its thrust, This point to which we wish to move it may be a point fixed with respect to the earth, i-t may be e point fixed in some inertial space or it rrlay be a point which moves according to an arbitrary physical law. The rocket will usually be moving initially and will be subject to the a celeration of gravity and possibly other specified accelerations. Then the displacement of the rocket with respect to the point is given by (7.1) f = - where X is the positio vector of the point in an arbitrarily chosen rectangular coordinate system wh::ose axes are not rotating ernd x is the position vector 6f the craft. Now we cAn write this as aPae $1

AERONAUTICALE RESESARCGH CENTER - UNIVERSITY -OF MICHIGAN Report No UMIM 19 19 d t -, 77 dt (7,2) = - dt + t G d where a is the acceleration due to thrust, $ is the vector of 40 initial relltive position, 4L is the vector of ilni -,' l reltive| velo-ity, -(t) is the cceleration to which -thie crFft is subject and I(t) is the acceleretion to which the trrg-et point is subject. The basic assumption of this metih od is that tle vector of displacerients d.e to extraneous forces (7.3) ff; a dt2 ff dt 2 does not de.eend upon the thriust aplied to th'e craf-t. e sh1all rev-ew this assurmnption in the next section. Def:CiL tion. Wie s.'all say that homin7l is effected if there is a time t2 > 0 such. tilh-t (7.4) 4(t2) = If we delnote boy- asrd S the vectors s= f j t dt s =5 + t ( t - ) dt 2, Page 32._

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF;MICHIGAN Report No UMM 19 then we can express (7.6) 5 = s-w We have expressed 4 in terms of two vect'ors, one of whiich depends only upon conditions specified by the target point and the other depends only upon thrust applied to the rocket. The definition of homing can be written (7.7) 4(t2) = (t2 Now for two vectors to be equal the first condition is thant their magnitudes be equal: (7.8) s = s, whe re s = I I S = ISI. But S is a vector known as a function of time. HIence we can plot S as a function of time. WJe can also reduce it to the dimensioniess form S3t as a function of ti- where S/(/r), t- r t, as in Section III. Now consider thrust fixed in direction. Then s is a given function of the burning time Et and of time t. IVWe can!We assume that r is not zero; if it is zero, then another parameter needs be chosen. Page 33

-AERONAUTICAL RES.E ARCH CENTER' UNIVERSITY.OF MICHIGAN Report No UMM 19 draw a graph of s as a function of t1 and t or of se-* as a function of t and t —. Consider the latter. We will have a set of curves exactly as in Figure III except that in general S* —(t-*:-) is not Aa hyperbola. All points such that S-:-(t*) lies below the curve of maximum performance for the rocket represent possible homing times. For any chosen homing time we read the burning time, represented by t-*:-, the abscissa at the point of tangency to the curve for s:-(t'-). The remaining problem is to determine the direction to apply thrust. The direction cosines are the direction cosines of S(t2) where t2 is the homing time chosen, namely X9 (t)x ) S (7.9) \(t2) = S(t2) A,(t2), V (t2) = Sz S2 S2 S2 where Sx, S and Sz are the components of S.. Paths of minimum fuel consumption can be determined in the same manner as before. If the S_-curve has an oscillatory character there may be several relative minima. Page 34

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMMI 19 VIII. DISCUSSION AND LIMITATIONS OF THE METHOD Let us consider some cases where the method applies. First there is the problem of homing against a target in free flight. W;e assumed that G = g. So long as G - g is an acceleration vector negligible with respect to other accelerations this is a valid assumption. Second, there is the problem of sending the craft to a specified point in space. For this case G - g is the gravitational a celeration on the rocket. Within the limits that we can estimate this as a function of time, this method is exact, ignoring aerodynamic forces. Third, the problem of homing against a target which follows a dodging course that can be predicted. Aga-in S becomes a known vector function of time. Let us consider some cases where it does not apply. A First, there is the dodging target whose maneuvers are based on intelligence about the pursuing craft. The vector S is no longer a function of time only since it depends upon the maneuvers of the craft. Page 35

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 19 Second, if the homing time is large, the displacement (G - g) dt2 may not permit accurate evaluation.. Third, the time required for rotation may be appreciable and require consideration. The condition that jf (G - g) dt2 can be computed as an explicit function of time is important for three reasons., First, if it is satisfied, the paths determined as above are the best paths that can be found from the point of view of obtaining maximum correction for given fuel consumption, and conversely, for obtaining a definite correction with minimum fuel consumption. Second, if it is satisfied, the problem of homing is reduced to a problem in algebra and the determination of paths of minimum fuel consumption, if they exist, is reduced to a problem in calculus with a direct graphic solution. Third, if it is not satisfied, the homing problem is very difficult to solve. One feature which can hardly be overemphasized is the gain, in reducing to dimensionless terms. In that way, a single grid covers all cases and the number of significant variables is reduced to a minimum. Page 36....

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 19 Of course, if a detailed analysis of a single craft is to be made, it may be more satisfactory to work with s, S, t, etc. The graph for s vs. t can be drawn up exactly as the curve for s*'r vs. t*':- was; If the curve for si is available, the curve for, s can be obtained directly from it; indeed the same curve can be used by altering the scales on the axes. All operations are performed as before except that times, distances, etc., are read in seconds, feet, etc., and require no conversion. Page 37

AERONAUTICAL RE.SE.ARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 19 IX. SOME APPLICATIONS OF THE METHOD TO PROBLEMS OF DESIGN The method -cn be used to get the answer to several design problems. As an example, assume that a set of target conditions is chosen and we wish to determine some craft specifications for homing. Assume that the specific impulse is givenl Now choose, as a first approximation a value of rQ that seems reasonable. Draw the graph of Si on the grid of Figure I. There are three possibilities. If the value of ro is well chosen, a figure is obtained similar to Figures IX.a, or IX.b. If ro is chosen too small, the resulting figure will look like Figure IX.c. If ro is chosen too large, the figure will look like Figure IX.d; the grid will appear dwarfed beside the hyperbola (this corresponds to thrust too large). It is best to choose a new value of ro in the latter cases so that the resultant figure is similar to IX.a. Otherwise, one of the 1The specific impulse depends principally upon the fuel for a well-designed craft,............ Page 38

AERONA.UTICAL RESEARGCH CENTER -UNIVERSITY OF MICHIGAN s~~~~~~~~~~~ S *~~~~~~~~~~~~~~~~~~~j *.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ t4 Fig. I a Fig.X b r3 i.n the proper range l f in.the proper range * * I " 5 ~ Fig. El c Fig. 33 d o too small l | too large Fig. To Grids for Various Values of rb and Fixed Target Conditions

AERONAUTICAL RE-SEARCH CENTER - UNIVERSITY OF MICHIGAN " Para lel to OPp to10 0 tio mI-2 t Fig.X Determination of r for Interception and of Lower Bound to Fuel Consumption.

AERONAUTICAL RESEARC H CENTER - UNIVERSITY X OF MICHIGAN Report No UMM 19 two figures,' the grid or the hyperbola, is s'o small compa red:to the other that interpretations are difficult. Two or three trials will five a reasonable value for ro. Then one may refine the choice oftro in various ways. First, let us show that for a chosen homing time, we can pick the minimum value ofQ that will cause homing at a selected homing time. This value is the value for burning all the way. Assume that we have the graph in a form like Figure IX.a, see Figure X., Choose any value of t*s- (or of t2). Designate the |point (t2*',S*(t*) ) as P2. Draw the line OPS. Designate the,on _( as,.OP point where it cuts the curve of sic(ti*) by P' Designate thel |abscissa of PL2 by t12*-. Then the burning rate for homing at time t2*- by burning all the way is t1 (9.1)'o't r. 0 t~.3 where r is the value for which the curve was originally drawn. It is a matter of direct substitution into the equations of Section II, Article C to show this. We have from Figure X (9.2) S(-(t2*-) S s2(t12) Q 2' since 0Q12P12 and 0Q2Pt are similar triangles. Page 39

AERONAUTICAL, RESEARCH RCENTER~ -UNIVERSITY OF MICHIGAN Report No UMMN 19 ~~C C ~~~~ ~~~~t2 t2 Now S(it2) -= S* (t2*),, s (t.) s: ( 12), and - 2 = t r ro 2" t=2 So equation (9.2) becomes (9. 3) S(t2) -= $(t2) * In- Article II.C. we saw that the burned fuel ratio' or two craft satisfied - the' relation (S. 12) p(t) = r(kt), Hence, letting k tk 2/t2 we ob~tain the desired result, (9.4) a-(tz) = i s(t12) tt2 - S(t2)1 if (9.5):o it t12 A rate of burning less than o cannot effect interception at the selected time and any rate of burning greater than o will effect homing at the desired time if t_ is properly chose~n. Fro-m equation (2..12) above we odtain the rel-tion (t2).= r(kt2) r(tl2), or (9.6) P(t2) = tl2:* Page:e 40.,

AERONNAUTICAL RESE EARCH CENTER I UNIVERSITY OF MICHIGAN Report No UMM 19 Hence tl2~-*- is the unit fuel consumption for burning all the way. In the illustration t2* was greater than t12*-. This was irrelevant to the: method and the result. However it is seen that certain points on the hyperbola, those for which S*-/t*- > 1, have no corresponding point P12 on the s —curve. For these points it is simply not possible to home by burning all the way. We see that for r a constant s ( t) < t f o r t < 1 lim s*. = lim [t"b + (1 - t*s-) In (1 - t-)]: (9.7) t,-I1 t-1 l as indicated on the graph. This gives the result (9.8) < c for r < 1; t in other words, the average velocity during burning is less than c, the eff ective gas velocity. Another quantity which may be rxead directly from the graph is the fuel consumption corresponding to burning in an imDpulse. In Figure X, draw the line (shown dotted) parallel to OP2 and tangent to the s-: —curve. The abscissa t-oU of the point of tangency gives the value of r corresponding to an impulse. This can be shown by considerations on velocity similar Page 41

:AEIRONAUTICAL RESEARCH CENTER UNIVERSITY OF. MICHIGAN Report No UMM 19 to those carried out on distance in the previous paragraphs, using the similarity considerations given in Section II. This value too* is the greatest lower bound to fuel consumptionl for the homing time t2.A Fuel consumption is monotonic increasing with burning time for similar burning. Hence we have bounds to the fuel consumption for homing at the selected time t2 t - o* r - t It If the amount of fuel is designated, this will show whether or not homing can be effected. The method of solution is also applicable to the problem of homing with minimum acceleration. A memorandum is being prepared on this. 1This was shown in detail in UMM-18. Page 42

AERONAUTICAL RESEARCH- CENTER-~ UNIVERSITY OF MICHIGAN Report No UMM 19 X. TWO THEOREMS ON HOMING Some interesting results follow from simple consid- | eration of the hyperbolas and the grids. W THEOREM Ii. If the burnt fuel ratio r can exceed r" = 1 - e c then homing can always be effected. Proof. The asymptote to the hyperbola will have slope W W:- =- and the hyperbola approaches it arbitrarily closely as t-i- becomes infinite. Since r > r", the curve of maximum per-,L formance has a slope greater than W*,. Hence for t* sufficiently great there are points on the hyperbola below those on the curve of maximum performance and these represent possible homing paths. THEOREM II. For an outgoing target the lovwer bound to the burnt fuel ratio is r" = 1 - e c Proof. For an outgoing target, the center of the hyperbola is on the t*i —axis either at or to the left of the 1These theorems were stated in UiM —18 as Corollaries VI..2 and VI.3 and the reference given is to this proof. Page 43

AERONAUTICAL RESE/AR.CH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 19 origin. If r < r" then the curve of maximum performance cannot touch the hyperbola, hence homing is not effected. By Theorem I, if r > r" homing can be effected. It is interesting that this Alimit does not depend upon the manner of burning; that is, upon the particular thrust function. We saw that the existence of paths of minimum fuel consumption depended upon the rate of burning. Page 44

A E:RO WITwI^CAL RES'E~ARCH CEENTER ~UNIVERSITY OF MICHIGAN Report No UMM 19 DISTRIBUTION Distribution of this report is made in. accordance with ANAF-GM Mailing List No. 6 dated August 1948, including Part A, Part C, and Part DA. "I~~ - Page 45

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