Homing in a Vacuum with Minimum Fuel Consumption EXTERNAL MEMORANDUM NO. 18 Project MX-794 (AAF Contract W33-038 ac-14222) *. ~ ":-.'.' Prepared by; " @; ofF. D. Faulkner' Approved'by'. C" ~ *'- ~:;' ~' -'~ Supervisor, Mathematics Group February 1, 1949 ~~~? ~ ~ ~ ~ Suerisr Mahmais ru Ferur 1,1949~

:}t1 AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN.I ~~q Report No UMM 18 Page ii

~NGINEERING LIBRARY AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 TABLE OF CONTENTS Section Page lI INTRODUCTION AND SUMMARY 1 II THE BASIC EQUATIONS AND FORMULAS 13 III THE CONDITION OF HOMIING 21 IV THE FIRST FUNDAMENTAL PRINCIPLE OF HOMING WITH MINIMUM FUEL CONSUMPTION 30 V THE SECOND FUNDAMENTAL PRINCIPLE OF HOMING WITH MINIMUM FUEL CONSUMPTION 37 VI THE LOUVER BOUND TO FUEL CONSUMPTION 46 VII THE NECESSARY AND SUFFICIENT CONDITIONS FOR A PATH OF MINIMUM FUEL CONSUMPTION 53 VIII FURTHER PROPERTIES OF PATHS OF MINIMUM FUEL CONSUMPTION 61 IX SPECIAL MINIMUM PATHS 65 APPENDIX 74''' Page iii

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN.r...~~ Report No UMM 18 ILLUSTRATIONS Figure Page I Homing Paths.,.... 32 II Velocity Curves for Accelerations a and a". 38 III Velocity Curves............ 41 IV Family of Velocities due to Impulse which Will Cause Interception... 47 V Conditions for a Minimum Path... 54 Page iv

AERONAUTIOAL RESEARCH CENTER - UNIVERSITY.OF MICHIGAN Report No UMM 18 I. INTRODUCTION AND SUMMARY The-problem treated here is that of homing with a point, mass. It is assumed that all action takes place in an isolated system such that there are no adjacent bodies to "push" against, no aerodynamic forces, etc.; the only outside forces are body forces, such as gravity. The homing particle, a rocket, hereafter called the craft, will be subjected to en extra force, called thrust, gained by emitting part of its own mass. The impulse, the time integral of the magnitude of the thrust, is assumed to be proportional to the mass of the consumed part, called the fuel. A second point mass, henceforth called the target, is assumed to be in the neighborhood of the first, near enough so that the body forces are the same on each. By homing is meant the application of thrust to the craft in such a manner that its position shall at some time,,-celled the homing time, This assumpltion is not necessary to most of the argument which follows; Iost of the discussion holds under more general conditions. Since the above case is the important one for rocketry, we limit ourselves to it. ~as"- - -— Page 1

AERONAUTIGAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 coincide with the position of the target. The motion of the craft during homing will describe a curve called the homing path. The homing problem arises in the following manner. Information will be received that a missile, the target, is on its way and approximate information as to its path will be given. The craft will be launched to go to some point on the anticipated path. As subsequent, more accurate information is received, it will be seen that the craft is going to miss the target. The question then arises; is it possible to apply the available thrust of the craft in such a manner that homing can be effected? We would like to know the answer to the following questions: 1. At what times (if any) is homing possible? 2. For a chosen homing time, what is the fuel requirement? 3. What is the best way to apply thrust to cause homing at a given time? 4. What is the least amount of fuel required to cause homing? The first two questions are treated in detail in a second report. The last questions are important because homing will probably be effected not in a single stage, but as a 1UMM-19, issued simultaneously with this report. Page 2

AERONAUTIOAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 series of corrections, each based on more accurate informntion than the previous. Only a limited amount of fuel will be available, and we wish to gain the utmost in correction from it. The study of the above questions brings up the problems of minimum paths. In this report we define: a minimum path is a homing path that requires less fuel than any other allowable1 homing path. In general, paths in the neighborhood of minimum paths require but little more fuel than the minimum paths. Let r be the burned fuel ratio, that is, the ratio of the weight of fuel burned to the initial gross weight. Let r" be the upper critical fuel ratio2, defined as the amount of fuel necessary to accelerate the crpft from rest to the initial relative velocity. If r can exceed r", then homing is slways possible for a suitably chosen path. But if r < r" then homing is possible only for minimum paths and for paths near the minimum path. This is an extremely important property. For project Wizard the burned fuel ratio in the homing stage is less than the probable upper An allowable path is a path which the craft is able to follow. This path must not, for example, require greater thrust than the craft can exert. There is also e lower critical burned fuel ratio to be taken up later. These quantities are defined more explicitly in the text. Page 3

AERONAUTIGAIL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 critical ratio; hence we are required to use paths near minimum paths. The proof of this property falls naturally into a later report and is not given here. It may also be asked: what is the information needed to completely specify the homing problem? The problem is completely determined if one knows: 1. The relative position of the target with respect to the craft initially. 2. The relative velocity of the target with respect to the craft initially2. 3. The performance of the craft; that is, its thrust-to-weight ratio, its rate of fuel consumption per unit weight, and the amount of fuel available per unit weight3. The first two of these will sometimes be referred to as the initial conditions. Homing may in many cases be achieved with less information, but the problem is indeterminate in general. Because of the importance of minimum paths and the fact that a minimum path is possible only for incoming targets, emphasis will be placed on incoming targets. A target is said to be incoming if the distance between target and craft is decreasing The upper critical fuel ratio is a function of the situation with which the craft is confronted. 2It should be noted that the actual velocity of the craft (and the target) are not required. This is in contrast to the usual homing problem in air. 3Note that these are all ratios. Page 4

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 initially. The analysis of homing holds for all tsrget situ-| ations except where specifically mentioned. Insofar as is possible, dimensionless terms have been used., For example, the mass of a rocket is unimportant in itself. The significant quantity is the ratio of the mass of fuel to the mass of the craft. This reduces the number of variables required to state a problem.Before starting on the problem of homing, some formulas of rocket motion are developed. It is shown also that the integral tdt is monotonic with respect to fuel consumption; T is the thrust vector and M is the mass of the craft. This result allows a simple attack on the problem of minimizing fuel consumption. The difference between the problem taken up in this report and the one in UMM-2 is this; in UMiM-2 the fina'l specifications of the craft are assumed- The attack here is the one which must be treated in the design of equipment for the field where it is necessary to consider the craft as it exists at the instant that homing starts; the initial craft, not the final craft, is specified.'a~-*~ 4 ~Page 5

AERONAUTIGAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 In general, the problem is simpler for this case: it is easier to solve the equations when all values are given initially than when some are given initially, and some are given at the (unknown) time corresponding to end of burning. The coo/rdinate system is chosen so that the origin describes the path that the craft would follow with respect to the earth if no correction were applied. In this coordinate set the target travels in a straight line with constant speed, and the entire motion of the craft is due to 1thA thrust. This, in effect, achieves a separation of variables; the initial conditions determine the target path, and the thrust determines the craft path. This choice of coordinates makes most of the theorems on minimum paths "obvious". An appendix is included to show that the results are independent of the choice of coordinate system, and to show how to transform from one system to another-. To eliminate Coriolis forces and centrifugal forces, we consider only coordinate sets whose axes are not rotating in inertial space'. That is, they are fixed in direction like the axis of a free gyroscope. Strictly speaking, the earth does not afford such a system naturally because of its rotation. For a more detailed discussion of coordinate systems see G.. Joos, Theoretical Physics, particularly Chapter 10. He calls a coordinate set such as that above an inertial frame,. that is, one in which the Law of Inertia (NeWton's Law) holds. Page 6

AE;RONAUTIGAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM1 18 The outstanding characteristic of the entire problem is its simplicity. It is entirely a problem in relative motion, since gravity, etc., act alike on target and craft. In this memorandum we have not taken up the problem of finding minimum paths. Fortunately they can be determined easily and quickly. The method of solution has so many other interesting and useful properties that it is put out as a separate report. As a final word I would say that the philosophy of this report is that expressed by von Neumann and Morgensternml "Economists frequently point to much larger, more'burning' questions, and brush aside everything which prevents them from making statements about these. The experience of more advanced sciences, for example physics, indicates that this impatience merely delays progress, including that of the treatment of the'burning' questions." It is our hope than an exact and painstaking analysis of the component problems will be valuable in treating the overall problem. For this reason, this report is somewhat more detailed than may seem necessary. To simplify the reading, the most important conclusions are drawn up as theorems and those J. von Neumann and 0. MVorgenstern, The Theory of Games and Economic Behavior, 2nd Edition (1947), 7. Page 7

AERONAUTIOGAL RESEARCH CE.NTER -UNIVERSITY OF MICHIGAN Report No UMM 18 theorems vital to the arguments which follow are indicated by an asterisk (*-). Conclusions 1. For the most efficient use of fuel, the thrust must be fixed in direction during homing. This direction depends only upon the initial conditions and the homing time. Any component of thrust not in this direction is wasteful of fuel. 2. For the most efficient use of fuel in homing, one should apply full thrust in the proper direction as soon as the error is known until the error is reduced to zero, then coast in to interception. An impulse represents the ideal way to burn fuel.. Any delay in starting wastes fuel. 3. The problem of homing, and particularly the problem of minimum paths, can be reduced 1An impulse is understood to be similar to that defined by H. Lamb, Hydrodynamics, Article 11, as the product of force and time, Ft where F - oo, t - 0, and the product remains finite, giving a jump in velocity. Account must be taken of the variation of mass of a rocket during the impulse. Physically an impulse is not possible, but certain properties of the case of an impulse are approximated very closely if burning time t ~<< t2 the homing time. Page 8

AER ONAUTICAL RESEA2RCH CE-NTER - UNIVERSITY OF MICHIGAN Report No UMM 18 to a two-dimensional problem by a proper choice of coordinate systems. For minimum paths, it can be reduced to a one-dimensional problem as a result of conclusion (1). 4w The lower bound to fuel consumptionl would be obtained by applying thrust as an impulse at 900 to the initial line of sight if the target is incoming. This is an immediate consequence of conclusions (1), (2), (3), and the equations of homing, There is no possible way for any rocket (with the same specific impulse) to home with less fuel. 5. There are three types of initial target conditions for a given rocket: a. Definitely incoming: here the target is incoming, the relative velocity is small enough, the initial relative range is great enough, and the acceleration due to thrust of the craft is great enough that a minimum path is possible. This is the lower critical burned fuel ratio referred to in an earlier footnote. Page 9

AERONAUTIGAJL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 b.. Essentially outgoing: in this case the target is either outgoing, or is incoming but is so close in when homing starts that the craft cannot move to cause interception before the target overshoots. c. Impossible: where the given craft cannot intercept the target (when r < r" and a minimum path does not exist). By delaying the start of homing or by poor maneuvering, a type (a) target may become a type (b) or a type (c) target. 6. For a definitely incoming target (and a finite burning time) the minimum path satisfies the following relations: a. The fuel consumption is less than that required to accelerate the craft to the initial relative velocity. b. The direction of application of thrust is greater than 90~ from the initial line of sight. 7. For an essentially outgoing target there is no minimum path. The lower bound to fuel consumption is given by the quantity of fuel, necessary to accelerate the craft to a........''..... Page 10.~~.~.

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMNM 18 velocity equal to the initial relative velocity. This lower bound corresponds to an infinite homing time; more fuel is needed for the practical case. 8. The necessary and sufficient conditions for a given homing path to be a minimum path are these; a. Thrust must be applied as indicated in conclusions (1) and (2). b. The burning time t1 must be less than the homing time t2. c. The thrust vector must be perpendicular'to the vector of relative position at the end of burning. 9. The necessary and sufficient conditions for homing to occur without further thrust, are that the vector of relative position be parallel to the vector of relative velocity and opposite in sense. 10. There are other cases of minimum fuel consumption. If the acceleration is constant during homing, the constant being chosen a function of t to cause homing, the lowest fuel consumption occurs if thrust is applied at 900 from the initial line of sight. The same is true if thrust is a constant all during the homeing flight, the Page.11 ~

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 constant value being properly chosen to effect homing and we have seen (conclusion 4) that it is also true if the thrust is applied as an impulse. These are all examples of a more general case; if the velocity due to thrust can be expressed in the form v = f(A,t/t2) where f is an increasing function of A for any fixed value of t/t2, then minimum fuel consumption is attained when thrust is applied at 90 from the initial line of sight. It is shown that the impulse corresponds to the case in which f(A,t/t2) A, constant acceleration to the case in which f(A,t/t2) At/t2 and constant thrust to the case in which f(A,t/t2) -c ln(l - At/t2). In general these are not true minima but represent relative minima subject to the above restrictions. The minimum referred to in paragraph 8 above is a relative minimum. There are some cases when there are two relative minima; the smaller of these is also an absolute minimum. In one other case there is a single isolated path which requires less fuel than the relative minimum. With these exceptions the relative minimum is also an absolute minimum; there is no possible way that the given craft can home with less fuel in the given situationl. l See UMM-19, p 20. Page 12

AERONAUTIOAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 II. THE BASIC EQUATIONS AND FORMULAS A. General Formulas In this section most of the basic formulas and equations needed later are developed, including the equations for velocity and distance, and the fundamental equations expressing fuel consumption in terms of acceleration due to thrust. The equation of motion of a rocket with no outside forces, derived from Newton's second law, is (2.1) M(t) d -T dt where M is the mass, u is the velocity vector, and T is the thrust force, of magnitude T. The thrust is gained by the rocket by the ejection of part of its own mass, the fuel, and if c is the effective velocity of the emitted particles, here considered constant, then (2.2) T = -M. 1The effective gas velocity c is actually a function of many variables including rate of burning and outside or "atmospheric" Page 13

AERONAUTIOAL RESE;ARCH CENTER -UNIVERSITY OF MICHIGAN Report No IUMM 18 We shall use the dot over a variable to indicate its time derivative. We shall use subscripts o, i, a, to denote values of variables at time to = 0, t1, t2 where to 2 t1 - t2. Let m be the mass of fuel consumed m = No - M(t) and define the burned fuel ratio, r, m r - Mo Then it follows that T = MOcr and M = M ( r). We can now write equation (2.1) in the form (-.) i=.: r. 1 -r Here a is the acceleration vector, and _ is the unit orientation vector of the thrust. We see that only the ratios of masses appear in this equation. It has the scalar form pressure. Engineers frequently use the term "specific impulse" denoted by "I", defined by the relation T I t h It follows from the definitions that Ig = c. Page 14

AERONAUTIGACL RESEARH CENTER - UNIVERSITY OF MICHIGAN,.....- -Report'No UMM 18 dI cr (2.4) = -r = a = a ay2 a2 dttl 1 -r r where a a. az ar where a, a az are the components of the vector of acceleration due to thrust along three mutually perpendicular axes, x, Z, z' We can integrate equation (2.4) with respect to time to obtain Ft ___ __ (2.5) J.a + ay + az dt = - c in (1 - r) This is the basic equation connecting the. burned fuel ratio with the acceleration due to thrust. Since r is the ratio of the mass of fuel consumed to the initial mass and In (1 - r) is a monotonic function of r, the problem of determining paths of minimum fuel consumption may be reduced to an equivalent problem of minimizing the integral in equation (2.5). Because of its importance we.state this as a theorem. THEOREM I. To minimize the fuel consumption required for homing it is necessary and sufficient to follow a path such that t in ra is mim eit hm. the integral j a dt is a minimum, where t2 is the homing time. Page 15 -

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN "I~ ~Report No UMM 18 Equations of particular interest are those for velocity and for distance. These are obtained from equation (2.3) by integration. t t (2.6) = a dt = dt 1 - r and (2.7) s u dt = a(-) d) dr J0 J0 J0 if the rocket starts at the origin with zero velocity. The last equation can also be written t (2.8') - = (t -) a(T) dr J0 = t a (r) dT- Ter (r) d r, J0 J0 or (2.8) s = t-s obtained by changing the order of integration and integrating, This defines 1'. Equation (2.8') is useful since it expresses distance as a single integral. Page 16 -

AERONAAUTICAL RESEARCH CENTER-UNIVERSITY OF MICHIGAN Report No UTIMM 18 An important case is the one when thrust is fixed in direction. Equation (2.6) can then be integrated to give the velocity, whose magnitude is (2.9) q = c in (1 - r) This equation allows a second expression of Theorem I. We shall make frequent use of this form so we state it as a second theorem. THIEOREM II. If one consideres only paths for which thrust is fixed in direction, then to minimize fuel consumption it is necessary and sufficient to minimize the velocity acquired from thrus t. This follows from relations (2.5) and (2.9) q = - c in ( - r) ft2a -=| a dt. We see that the velocity in this case depends only on the amount of fuel burned and not on the rate. of burning. The relation for distance (2.7) has several forms, among them (2.10) s = - c In (1 - r) dt Page 17

AERONAUTIOAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UIIM 18 and t s = c I (t - r) r(r) dr; 1 - r(r) these expressions, however, cannot be integrated until r is specified as a function of time. B. Thrust Constant and Fixed in Direction The case when the thrust vector is constant (fixed in direction and in magnitude) is of particular interest for two reasons - 1. It affords a good approximation to iiany rocket problems. 2.'It is frequently the simplest from the engineering and design viewpoint. This case is characterized by a constant rate of burning; that is, r = constant = r 0 (2.11) r = rot. The formula for velocity (2.9) becomes (2.12) q = -c In (1 - ot-) during burning. Let us define the burning time tL by Page 18 ---. I

AERONAUTIOAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 the relations r = ro = constant for t < tl, (2.13) r - 0 for t > t. Then we have (2,14) q = - c In (1 - t) for all time t such that t - tl Since r is now a specified function of time, we can integrate equation (2.10) to get distance, (2.15) s = [r + (1 -r) in (1 -r) while burning is continuing. This has the alternate form (2.16) = ct [+ ( 1) I n (1 - t)] r(ot If t_ is the burning time defined by the relation (2.13), the formula for distance has the form (2.17) s(t,ti) = (t - tl)ql + s1 = q1t - s1'Page 19

AERONAUTIOAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 for time t - tl, where qi = q(tl) sl = s(.tl) sl' = qitl - S1. We can combine relations (2.11), (2.14), and (2.16) and substitute into equation (2.17) to get (2.18) s(t,tl) = -ct ln(l - rtl) + ctl [l + I ln(l - roti)] r~tl If we take: t = t during burning, then equation (2.18) includes equation (2.16). Equations (2.16) and (2,18) completely describe the one-dimensional motion of a rocket whose motion is due to prescribed constant thrust for a specified time. Page 20

AERONAUTICAAL RESEARCH CENTER UNIVERSITY OF MICHIGAN Report No UVMM 18 III. THE CONDITION OF HOMING1 Let us denote the position vector of the craft by x, whose components will be x, A, and z; the position vector of the target by X = (X,Y,Z); and the vector of relptive position by = (,rr ) = X - x. ~We shall consider only rectangular cartesian coordinate sets whose axes are not rotating in inertial space except where it is specifically stated otherwise. It is shown in the appendix that (3.1) = o + o t - a dt2 The vector a is the acceleration of the craft due to thrust, a =- T/M. By ft a dt2 is meant f I a(r) do o; d we shall use this notation throughout when no confusion is likely to arise. In the introduction we defined homing: by homing is meant the application of thrust to the craft in such a manner that its position will at some time coincide with that of the target. Page 21

AERONAUTIGOAL RESEARCH CENTER UNIVERSITY OF MICHIGAN Report No UTMM 18 We can interchange the order of integration in equation (3-.1) to get (3.2) a dt2 = (t -) (rT) dr tu(t) - x' (t). We may consider u as the velocity due to thrust; x' t a. dt has the dimensions of a displacement. Mathematically, the definition of homing is the following: (3.3') (t2) = 0, for some t2 0; tha.t is, (353.) to t+ ot2 - t22 + x2 = 0, for some t2 >-, where u2 = u(t2), etc. Since a = IlI is zero for time b > t1, the burning time, we have the relations UI = u(t) = u = constant, (s.4) = x'(t) = x = constant, for tltt t2 Page 22

AERONAUTIGAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 From equations (3.1), (3.2), and (3.4) we get the relation 4 (t) = to- TU == constant, for t t - t; the relative velocity is constant for time greater than to. In the same way we can prove the lemma. LEMMA., The relative velocity 4 is constant during any time interval durin, which thrust is zero. We can now prove the Theorem of Homing..sTHEOREM III. Thbe necessary and sufficient conditions that homing occur without further corrective thrust are these: At the end of burning, either (1) the vector 4, of relative position must be parallel to and opposite in sense to the vector 41 of relative velocity; or (2) the vector of relative position must be the zero vector. Proof. We can write $ in the form (3.5) = - k 4- + A where A is a vector perpendicular to e, and k is a constant with the dimension of time., — i ~' V Page 23 _

AERONAUTIOAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 Now (3.6) 4 = t, for t > tl, by the lemma above. Hence we can integrate to get (3.7) - = (t - t )1, for t > t, or, combining equations (3.5), (3.6), and (3.7), (3.8) 4 = (t - tj - k) 41 + X. It is necessary and sufficient for 4 to be zero at some time tz >tl, theat A = O (3.9') k 0; that is, that ~l = - k &, (3.9) k = t - ti ~ 0; the theorem is proved. COROLLARY III. The vector of relative velocity is parallel and opposite in sense to the vector of relative position for all times t greater than the burning time t and less than the homing time t if the craft is on a homing course.......... Page 24 l

AERONIAUTIOAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UIJM 18 Proof. Substitute relation (3.9T) into equation (3.7). Expressions (3.9) are called the conditions of homingl. From them we see that the conditions k > 0 and k = 0 correspond respectively to Cases (1) and (2) of the theorem when burning time is greater than homing time and when homing equals burning time. The Case (2) is an extreme case, and we shall be primarily concerned with Case (1) when t2 > tl. We can write the conditions of homing as XO + Ut1 - uzti + xl' Yo + Vtl - vit + Yi' (3.10) U - uz V- v Zo + Wtl - lt + z - t Wt - W Here (XQ, Q, Z) is the initial vector of relative position, and (U, V, W) is the initial vector of relative velocity; the other terms have been defined. We preclude terms in (3.10) which have zero as both numerator and denominator. By subtracting t from all terms in relation (3.10), we obtain another form (3.11) XO + x' = Yo + Y1' = Zo + z1 t, < - t U - u V - v W - w 1Relation (3.3) is sometimes called the condition of homing, but it seems desirable to consider it a's the definition of homing. L ".~-"''~ -'Page 25

AERONAUTLIOAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 It is shown in the appendix that by a proper choice of orientation of axes, we can always choose our coordinate set so that Y = Z = O (3.12) W = 0 or so that Zo = 0 (3.12'.) U = W = O There is no real loss of generality if we consider the plane problem; for three dimensions, equations are replaced by systems of equations, and the algebra becomes laborious. In the next sections we shall see that the most important cases are the plane cases. Hence, we shall be concerned almost entirely with the plane problem. The conditions of homing then become Erelations (3.10) and (3.11)] (3.13) X0 + (U - u1) t1 + x1' _ (V - vi)t1 + y' U - uI V - vl (3.14) +. O' - t2 - ti.* U - U V - ^~ ~~~~~Page 26 ~

AERONAUTIGAL RESEARCH CE.NTER UNIVERSITY OF MICHIGAN Report No UMM 18 The conditions of homing always consist of two parts, an equality which states that the two vectors L and 4 are.parallel and an inequality which states they are opposite in sense. The equality of expression (3.14) becomes (3.15) (Xo + x,) (V- vI) - Y1' (U - u) = 0; this is called the equation of homing. A case of particular interest is the case when thrust is fixed in direction at an angle G_ with the x-axis.. If we let t q = a dt =- c n (1- r) 0 (3.16) t s = r a( r) dr then u q cos, v = q sin, (3.17) X1 = st cos e, y' = s' sin. The equation of homing (3.15) then has the form (3.18) XoV + s1' V cos G - qlXosin G - Usl' sin G = 0 Pa- e 27

AERONAUTIGAL RESEARCH CENTER UNIVERSITY OF MICHIGAN Report No UMM 18 Other forms of the conditions of homing: it seems worthwhile to point out some other forms of the conditions of homing. If we denote the direction cosines of 4l by (XA,g,v), we get the conditions of homing as dl'l < 0, dt t = tl (3.19) (XU~,v) = (l,/,1, i), for t > t1. In two dimensions we can write this Ps dlJl < 0 dt (3.19') a = a. for t' t; here a = arctan (Y - y)/(X - x). They can also be expressed as (5.SO) _ _ 4A 1 = =0 (3. 20) 1. * 1 < 0, where the symbol ^ denotes the vector product. Still another form is the following: if we denote by M the angle between 4_ and ~_, the conditions of homing Page 28

AERONAUTIOAL RESEARCH CENTER- UNIVERSITY OF MICHIGAN Report No UMI 18 become (3.21) 4 = T. Numerous other expressions can be found, but they can all be reduced to expression (3.9). As stated previously, they always consist of two parts: an expression of the equation of homing and inequality. In equation (3.21) the inequality is implicit in the definition of.| Page 29

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 IV. THE FIRST FUNDAMENTAL PRINCIPLE OF HOMING WITH MINIMUM FUEL CONSUMPTION Consider the homing problem in the following form. Let the craft start at the origin end move only under its own thrust. Let the target be originally on the x-axis at a distance Xo from the craft and let it move with constant velocity whose components are U, V, O (zero). It is shown in the appendix how the coordinate set can always be chosen to put the problem into this form. The position of the craft is given by (4.1) x = a dt2 = (t- ) a (T) d, where a = T/ii is the acceleration vector resulting from thrust. The position of the target is given by X = Xo + Ut (4.2) Y = Vt Z = 0. Page 30

AEERONAUTICAL RESEkARCH CENTER -UNIVERSITY OF MICHIGAN Report No UTIM 18 The following analysis could be carried out for any coordinate system, but the above choice makes the results and the interpretation clear. By definition, homing is achieved if there is a time t2 such that t2 (t2 - t) ax dt = X + Ut2 t2 (4.3) (t2 - t) ay dt = Vt2, t2 (t2 - t) az dt = 0 The first fundamental principle of homing with minimumr fuel consumption is that thrust must be fixed in direction. We prove this as a theorem. *'THIEOREM IV. If one is given any homing path for which direction of thrust is varied, one can find a better path, and in this better path, thrust is fixed in direction. By a better path is meant a path such thet homin- is achieved at the same time, hence the same place, such that less fuel is required and such that neither the thrust nor the ~ ~ -~t~~~~~ ~Page 31

AEIRONAUTIOAL RESEARCH -CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 acceleration required exceeds that required by the given path at corresponding times. Proof. Assume that a homing path be given for which angle of thrust varies. The homing time t will be specified, and the functions ax, a, az, which describe the path. They. will satisfy equations (4.3),:(X0 +Ut2,Vt2), point of homing Given craft path Its projection on the xy - plane \ Better path to cause homing at the same M. (Xoraft~(OO,O) Y a Target position z time t=0 Figure I. Homing paths Let us rotate coordinates through the angle G = arctan Vt2 X + U t2 ~~Page 532 ~ ~' time t32

AERONAUTICAL RESEARCH CENTER UNIVERSITY OF MICHIGAN Report No UMM 18 so that the point of interception is on the new x"-axis (see Figure I). The result is now apparent from Figure I. It is required that the craft move from 0 to P under its own thrust. If thrust is applied with a component not along OP2, the craft will move off the line OP9 and miore thrust will be required later to bring it back onto the line. Both components were wasted; since thrust represents fuel, fuel was wasted. The straight line represents a better path. The analytic proof follows. We have x" = x cos 9 + y sin 9 (4.5) y" = -x sin 9 + y cos z =z with corresponding expressions for X, Y, Z and ax, ay, az. The conditions of homing (4.3) then become t2 (t - t) ax"t dt = X2 \/(Xo. Ut2)2 + (Vt2)2 t2 (t2 - t) ay" t = 0, (4.6) t2 (t - t) az" dt = 0,... ~. Page 33

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 and a and az" are not both identically zero since thrust is not fixed in direction. Now consider thrust applied so that the accelerations are as follows: a X"= — Vax 2 + a "T2 + a 112 -x y z (4.7) ay = 0 r" = c " = 0. z This set of accelerations satisfies the last two of equations (4.6) but not the first, since t2 t2 (4.8) (t2 - t) ax" dt > (t - t) ax" dt = X" 0 Now let us reduce the burning time as follows: choose the burning time ti" to satisfy the eouation (4.9) (t - t) ax" dx = X2 The accelerations (ax", >'z ") satisfy the conditions of homing with burning time t1" < ti the original burning time. But the thrust is identical in mLgnitude for the two cases for t < tl"; therefore the accelerations are the same Page 34

AERONAUTICAL RESEARCH CENTER UNIVERSITY OF MICHIGAN Report No UMM 18 in magnitude at corresponding times..We have decreased the fuel consumption by decreasing the burning1 tire. T'herefore we have found a better path, and in this better path, thrust is fixed in direction, Many conclusions can be drawn from this; let us suinmarize some of the more important ones. COROLLARY IV.1. For a minimum path, the angle of thrust is fixed. COROLLARY IV.2. The study of minimum paths is a plane problem. By our c'hoice of coordinates the target motion is constrained to a plane1. By the theorem above and by Corollary IV. 1, a = 0 for a minimum path. Therefore the craft motion is also confined to the xv-plane. COROLLARY IV.3. The study of minimum paths can be treated as a linear problem. By Corollary IV.1, thrust is fixed in direction for a minimum path. In the coordinate set used in this section The plane is moving with respect to the earth under the acceleration of the earth's gravitational field. The plane also rotates with respect to the earth with an angular velocity equal to the negative of the earth's angulpr velocity in inertial space. Page 35

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 a minimum path is a straight line. Hence we can drop subscripts in equation (4.9) and write the equation of homing as t2 ta (4.10) (tdt - t) a d = (a dt = X2; we treat t2 as a parameter and X2 is a function of t2. -COROLLARY IV.4. By a proper choice of coordinate set the problem of determining mrinimurn paths is reduced to the problem of effecting homing with minimum velocity. The integral to be minimized is (4.11) a dt = q2 - c ln (1 -r). -~~ ~ Page 36

AERONAUTICAL RESEARCH CENTER UNIVERSITY OF MICHIGAN Report No UMM 18 V. THE SECONID FUNDAM',EN ITTAL PRINCIPLE OF HOMING WITH M'INIMUM FUEL CONSUMPTION We have just seen that in order to minimize fuel consumption thrust must be fixed in direction; consequently the problem of minimum paths wes s -own to be s linear problem. For a chosen homing time, the problem now is to determine the best way to move along a stra iht line in inertial space. The equation of homing can be written (see the last paragreph of Section IV) tz (5.1) (t2 - t) a dt = D2 Here D2 = \(Xo + Ut)2 + (Vt2)2, and a is the magnitude of a, the vector of acceleration due to thrust. The thrust must be directed to accelerate the craft in the direction of G arctan Vt2/(Xo + Ut2). W!e must choose a to satisfy equation (5.1) and to minimize the integral Page 37 ~

AERONAUTICAIL RESEARCH CENTER — UNIVERSITY OF MICHIGAN Report No. UmI 18 2 (5.2) a dt = q2 in equation (4.11). *THEOREM V. If the homing time is chosen, the best way to apply thrust is this: apply full thrust in the proper direction until the right velocity is reached, then coast to interception. q q', for max. thrust q \ for thrust less than max. \^. q' t; t_________ Figure E. Velocity curves for acceleration a' and a" Proof. Consider a', the maximum acceleration which the thrust can effect; it is a function of time. Let q' be Page 38

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 the velocity defined if the acceleration is applied for a time tL', then the craft is allowed to coast to interception. Let q" be the velocity defined by any other acceleration a" (see Figure II). Now a" < a' for part of the time interval (0, tl'). Hence q"(tl') < q'(tl'). Now if qg is to cause homing at the time t_ it must satisfy the relation t2 q" dt = D dt2q = q' dt. But tl' tl' q" dt < q' dt; hence t2 t2 q' dt > q' dt = q2' (t2 - t1) Hence (5,3) qmax > q'max =' Page 39 ~ -~.

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN ~~r-~ ~~Report No UMNIMI 18 This is obvious in Figure II if we interpret acceleration as the slope, velocity as the ordinate, Cnd distance as the area under the curve. Therefore,'the peth defined by a' requires less fuel. This theorem has several interesting corollaries. COROLLARY V.1. Any delay in the start of homing is wasteful of fuel. This is a direct consequence of the theorem since the curve for q" lies below the above curve for q' when t < t1. Ience it must rise above it later as in Figure II. This represents greater final velocity, hence greater fuel consumption. COROLLARY V,2. Thurst applied as an impulse is most economical of fuel. Proof. If thrust is applied as an impulse, q reaches its maximum value q2 = D2/t2 at time t = 0 (see Figure III) so that the q-curve extends to tLe q-axis. Now if the thrust a" is finite, the curve for velocity q" defined by it lies belo-w the line q = D2/t2 in a neighborhood of t = 0. Since the a.:ea un-der each curve must be D, the latter curve must rise ov-r the line q = D2/t2 later; hence this path requires more fuel. Page 40

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMI 18 The analytic proof follows-. It consists of two parts, the proof that for an impulse q2 = D2/t2, and the proof that if the acceleration has an upper bound, then q2 > D2/t2, q -q" -Velocity curve when thrust is finite Velocity curve for an impulse at t0O 02= t, tP Figure III. Velocity curves We define the impulse as the limitingr case when the burning time approaches zerol. This implies that the acceleration becomes infinite. We characterize the other cases by finite accelerations, that is, by finite thrust. See footnote, p. 7. 2These characterizations seem to emphasize the important properties of the two cases. The two categories are not all-inclusive, however; for example, if the acceleration a be defined by ~~-~~~ ~Page 41

AERONAUTICAL RES EARCH CENTER UNIVERSITY OF MICHIGAN jp~~[~ ~Report No UMM 18 Let us expand equation (5.1) tl (5.1 ) q2t2 - t a dt = D2 From the results of the preceding chapter we consider only a > 0. We have then (5.4) q2 -D2/t2 Now.tl 0 - t a dt 3= sl') <- t | a dt, or (5.5) s^ < t-q, 3s = tjq2 al, sa constant, for t < 2 a = _ _D2 t, for t > a at (t^ - D) at2 then substitution into equation (5.1) shows that a satisfies the equation of homing for all values of al > D2/t2, including the case when aI becomes infinite; yet the burning time t4 is always equal to the homing time 2. Such systems do not seem practical and we shall not consider them further. Page 42 1

AERONAUTIGCAL RESE;ARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 Hence < D2 tiq2 (5.6) q2 +. t2 t2 If we combine (5.4) and (5.6) to get ( 5< < D) (5.7) ~ ^ qg = t2 t2 - tl it follows that D2 (5.8) lim q2= tf-O t2 The first part of the theorem is proved. We see from relations (5.5) and (5.8) that (5.9) lim s' = 0. tryO On the other hand, consider the case when the acceleration a is finite. Let a' be an upper bound to a. t Let q be a dt, and define ti" and q" by the relations tI (t2- t) a" dt = D2 (5.10) qt = a"t, for t < tl" ql = ai"t " (not al"t1).; Page 43

AER ONAUTICAL RESE.ARCH CENTER - UNIVERSITY OF MICHIGAN'~ ~~'~..Report No UIJM 18 Now t2 t2 q dt = q" dt ( = D2) But q q"t for t < tf so that t ~, q dt q" dt and t2 t2 q dt 2 q" dt 4- n g-d- if Since e t re aximurn value q2 of the integrand is greater than or equal to its everg,-e value, it follows that q2^ - q2 D2 q _ D2 D2.^ >,= > D A L-2a"tg Page 44

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 The last relations come directly from the definitions in relations (5.10). By Theorem II, the csse of the impulse requires less fuel. The proof is complete. There now remains but one problem in the determination of the lower bound to fuel consumption, the determination of the best homing ti xe. Thrt is taken up in the next section. Pag e 45

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 VI. THE LOWER BOUND TO FUEL CONSUMlPTION We have just seen that if the homing time is chosen, the best way to apply thrust is to make it as large as possible during the early rmoments of homing. This leads to the impulse as a limit as the burning time -t1 approaches zero. The only remsinin problem is to determine the proper angle of thrustl 9 for the impulse, or to choose the best hofming time since the two are connected by thle formula Vt2 ~)(6) tan 9 - X Ut2 We state the results as two theorems, one for incoaming targets (U < O) and the other for outgoing targets (U > 0). THEOREM VI A: For an incoming target the lower bound of fuel is obtained if thrust is applied as an impulse normal to the initial line of sight.!Since we are now working in two dimensions, a sing-,le angle will specify the direction of thrust: we call 9 the angle of t;hrus t. ~ Pae 46

AERONAUTIGAAI RESEARCH CENTER UNIVERSITY OF MICHIGAN Report No UMM 18 Proof. Assume the target is initially at the point (Xo,O) with velocity components (U,V) where U < 0 (see Figure IV). Consider an impulse such that it gives y \ \ Locus of velocity jump e(for q, a min) V U o raft Target (tAO) (t-o) Figure 3. Family of velocities due to impulse which will cause interception, a velocity jump of magnitude t making an angle e with the x-axis. The equation of homing (3.18) XoV + sl' V cos 0 - q1Xo sin G - Usl' sin 9 = 0 becomes (6.2) V - ql sin 9 = 0 since sl' = 0 for an impulse and Xo f 0. Page 47

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 We wish to minimize (6.2) q = sin 9. as a function of 9, This has the obvious minimum for 9 = 90~; (6.3) qlmin = V The proof is complete. The graphic proof is so simple and enlightening thqat wve include it, see Figure TV: consider an impulse such that the velocity jump ql satisfies the relation (6.2) qz sin G - V = 0 From the graph we see that ql is smallr for 9 = 90~ (and that all conditions of homing are satisfied since Yi = y1, Yi = Yi, 1- <10, X1 X- > 0). The corresponding fuel consumption is -. V (6.4) r' = 1 - e c tthis is the low/er bound to fuel consumption. It seems worthwhile at this point to review the:il chain of reasoning for this important result. We have shown: 1. If a homing path is given for which angle of thrust is varied, one can determine directly a better path for Page 48

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF.. MICHIGAN Report No UMM 18 which angle of thrust is constant.. Consequently a minimum path must have thrust fixed in direction. 2. If the angle of thrust is fixed, the best way to apply thrust is to apply thrust as high as possible erly-, then coast; the impulse is the ideal. 3. For an incoming target, the best direction for an impulse is at right anles to the initial line of sight. Thus, there is no possible way to homne which would require less fuel. For the out oing tearet, we have THEOREM VI B: Ag ainst an outgoing targt, the lower bound to fuel consumption is (6.5) r" 1 - e c where Q U +V Proof. Let us coimbine the inequality (6.6) cos ~ < cos G (see Figure IV) -with the equation of homing for an im-ulse (6.2). Tow q < 900, hence cos >- 0 Page 4 9

AERONAUTICAAL RESEARCH CENTER-UNIVERSITY OF MICHIGAN Report No UMM 18 We canl rewrite equation (6.2) as V (5 7) l (1 - cos2 G) With relation (6.6) t.:is gives the greatest lower bound (6.8) gib. (ql) = V /(1 - Gos2 i ) - Q and we can come arbitrarily:lose to this limit. The corresponding( lower bound to fuel cons-umption for an outgoing target is (6.9) r = - e c There are several interesting consequences. L EMMA. As the angle of thrust G approaches the a.ngle _ between the initial line of sight and the initial velocity vector, the hominl timre becomes infinite. We can solve equations (6.1) and (6.2) for t2 when thrust is fixed in direction X0 si, 9 (6.10) t2 Q sin (. - 9) Page 50

AERONAUTICAL RESE7ARCH CENTER - UNIVERSITY OF MICHIGAN Report No UM"I 18 The denominator goes to zero as 9 approaches Q and the numerator remains finite. From this we get an important corollary to the theorem. COROLLARY VI.1. For practical purposes, more fuel then the lower bound r" = 1 -e c is required for homing, against an outoin target. Proof. Thie limit is approached only as G ap.proaches. In this case the homing: time becomres infinite (equation 6.10): the target would have completed its mission before hornin-,, occurred. Two other corollaries can be proved. However, they involve considerable manipulation and they follow from other considerations as obvious conclusions. Hence we state them here and indicate'a proof. P different proof is given later COROLLARY VI.2. If thrust is not infinitesimal, then the lower bound to fuel consumption, r" 1 - e c for homing a;ainst an outgoin stargot is inde)endent of the thrust function. That is, if the thrust is finite, the lower bound approached as tL becomes infini' te does not de end upoo the thrus t unc i on. UTIjM-19, Section X. Pas;e 51

AEI0ONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN.........Report No UMM 18 Thpt can be proved by developing the relations D2 D2 XD + O t q 2 < + <t 2 t2 t2 - tl t2 - tl rand using th.e relation thaat as t2 becoomes infinite, t1 remains bounded if T is not infinitesimal. Q COROLLARY VI.3. If the fuel consumption exceeds r" = 1 - e then homing is always possible. Inspection will show that we have not used the relation' 90~ in the previous,)roof and that the results hold independent of q. The corollary follows. So we see that it is always possible to home if r > r". It is an.important property of minimumn paths that only for peths rear them- is it possible to home with less. 1These tw —o corollaries cen elso'e provId directl- from considerations on- the equation of homfng (3.18) XV + sl' V cos - ql Xosih G - s' U sin = 0 but the proof in~volves mrany details to im:ke it corplete. Page 52

AERONAUTICAL RESEARCH CENTER -- UNIVERSITY OF MICHIGAN Report No UMM 18 VII. THE NECESSARY AND SUFFICIENT CONDITIONS FOR A PATH OF MINIMUMiI FUEL CONTSUMPTION In this section are developed the necessary and sufficient conditions which a path must satisfy to be F path of minimum fuel consumption. The proof is long and detailed. Hence we state the principal result'at the beginning as a theorem and put other direct results in the next section. *THEOREM VII. The necessary and sufficient conditions for a' path to be a p th of minimum fuel consumption are: (1) The first two fundamental principles of homing with minimum fuel consumption (of Sections IV and V) must be satisfied; that is, (a) thrust must be fixed in direction and (b) thrust must be as large as possible during the early stage and then as small as possible, (2) It must satisfy the conditions of homing with tl < t2. Page 53

AERONAUTIOAI RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 (3) The vector of relative position at end of burning must be perpendicular to the thrust vector (see Figure V). y Qcos()- em) p osiiPo, target position (t tm) Im \m Tangentl ^Page 54 point Craft position (t~t^) V^ Xo em m-n+t(-Vcosem+ Usin m) ~ Craft X~ 90"+ e Target (x.1o) (t=O) (t=O Figured. Conditions for a minimum path proof.'Me have established thet for a chosen homing time, conditions (1) must be satisfied to minimize fuel consumption, The remaining problem is to establish conditions on

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 +, or what is equivalent, on 9 since they re connected by the relation (4.4) (7.1) g = arctan ~. Xo + Ut2 By the above principles, the unit burning rate r and the unit fuel consumption r are given functions of time for t < tj. Hence the acceleration (7.2) a = cr l -r is also a klnown function of time, We must determine the angle of thrust 9 to minimize fuel consumption. Consider the equation of homing (3.18), (7.3) XoV + s' V cos 9 - qXo sin 9 - UsI' sin 9 = 0 This is a function of the variables 3., P, and 9 (and of the known parameters and functions r(t), c, U, V and X ). *Hence we can write it as (7.4) F(s,q,9) = 0 Assume thrt r (or t.hat the thrust) is not zero for any time t < tj. Under the above assumrtions (thpt (t) is given and that thrust is fixed in direction) it is obvious from the definitions (a = cr), 1 - r Page 55

AER-ONAUTIGAL RESE:ARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 (7.5) ql = a dt, s (' t) a dt, 81 = t a dt, rl = dt, ^0 KJO d that q, 8' and l are all strictly increasing functions of t. Hence, implicit function relations are satisfied; all are determined as soon as one s chosen; and to inimize one of them is to minimize them all. We find it simplest to minimize'I is.. If homing is to be effected, equation (7.3) must be satisfied. Define the total partial derivative, (7.6) F aF + aF dsl'(ql) 0,1 s q )s3' dqj Then so long as 8F/bq1 l 0, equation (7.3) or (7.4) then defines | qg as an implicit function of 9; we can write DF aF F dq 0 DG. D9 bq dG since equation (7.4) must be treated as an identity in 9, q1(Q); hence - (7.7) dq _ aF. oF I d9e a9d oqI Paj:e 56

AERONAUTICAIL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 For our case we can (and will later) show thst FF/bql is bounded; hence, a first condition for q, to have a minimur is (7.8).F 0 be Substituting from (7.3), this becomes (7.9) sl' V sin G + qlXo cos 9 + Us,' cos 9 =.0 Now define a by the relation SI'V (7.10) tan a = qlXo + sl'U We can use the conditions of homing to show thCt, (7.11) si'V _ Vtl - sl sin qIXo + s1'U Xo + Ut1 - Si cos G Y1 - yi XI - Xi Hence a is the angle rmde by the vector of relstive position st the end of burning with the x-axis (see Figure V). From equation (7.9) we get (7.12) tan G - + U S.1 tV Comparing this with equation (7.10), we. see th.t for a minimum path it is necessary thst the vector of reletive position st Page 57

AERONAUTICATL RESEARCH CENTER-UNIVERSITY OF MICHIGAN Report No UbIM 18 end of burning be perpendicular to the angle of thrust. Conclusion (3) is proved. We must prove that the conditions of the theorem pre sufficient; that is, that equation (7,9) and the conditions of homing lead to a minimum, not to a maximum nor an inflection point when t1 < t2. WJe shall need the relation ds1' (7.13) d = dql We substitute this relation into equation (7.7) to get (7.14) dq -'_ q1Xo cos 9 + si'V sin 9 + si!U cos 9 dG Xo sin 9 + t1(-V cos 9 + U sin 9) Let us denote the values associated with a minimum path by subscript m. If we expand the numerator of the right hand side of equation (7.14) in powers of 9 - 9 we find that m is a simple zero, and the numerator is positive for 9 < G (9 near 9). Hence, taking into account the minus sign in front, we see that the denominator must be positive for a minimum; we must have (7.15) Xo sin 9m + ti(- V cos 9m + U sin Gm) > 0 This we can write as (7.16) X + > cot 9m VtPage 58

AERONAUTICAL RESEJARCH CENTER - UNIVERSITY OF MICHIGAN Report No -JMM 18 Now if we express the position of the target in polr form S =;/X2 + y2 (7.17) X Vt e = arctan = arctan Vt Xo + Ut we see that' is an increasing function of time (since V and Xo are positive). We can write (7.16) as (7.18) cot 6 (t ) > cot 0 (t2) He-nce tzm < t2m Inspection of the conditions of homing with thrust fixed in direction will show that thhe inequality, which is the condition that the vector of relptive positio' and relative velocity be opposite in sense, can be expressed as Hence the condition (7.16) is satisfied for all noming paths such tn-t t1 < t2. Hence we have proved th-t conclusion (2) is necessary and t at conicusions (2) and (3) are also sufficient. Let us sumnmarize the proof: the equations of motion when thrust is a s ecified function of timel, fixed in 1This specified function is the upper bound. Page 59

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 direction and not zero for t < t1 define q1, s3,, r end st as strictly increasing functions of one another; hence, t-here is an implicit function relation between any two and if conditions are chosen to minimize one, all are minimized. The equation of homing in turn defines qg as a function of G. Wle find dql/dG. For ql a minimum dql/dG must vanish. This is shown to be the condition that thrust be perpendicular to the vector of relative position at end of burning. The numerator, of dqi/dG has a simple zero so the second co ndition for a minimum is that the denominator be positive. This condition is satisfied by the conditions of homing whenever the burnin, time is less than the horming time. No real trouble is experienced if we do not assume a to be non-zero, since q, and s,' are still strictly increasing functions of one another. The proof is complete. Page 60

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UIMM 18 VIII. FURTHER PROPERTIES OF PATHS OF MINIIVUIVII FUEL CONSUMIViPTION The following properties of minimum paths can be deduced at once from t'he results of the last section'and are essentially corollaries to Theorem VII. Because of their importance we include them as theorems. TiEiOREM VIII.A. TIhe anile of thrust for a minimum path is greater than or equal to 900 (from the initiel line of sight). It is greater than 900 if thrust is finite and equal to 900 for an impulse. Proof. Let us eliminate a, between equations (7.3) and (7.9). We get S1T (8.1) cos 9m = X- o C oxo Gm is the angle of thrust for a minimum path. Now ftl s1 - | t a. dt > 0..... -Page 61 ~ - ~-.~P g

AERONAUTICAL RESEJARCH CENTER -UNIVERSITY OF MICHIGAN Report No UTJM 18 if thrust is finiteI. For the case of an impulse we saw that (5.9) lim s' = 0 tl-~ Hence 9G > 900 if thrust is finite (8.2) 9m = 900 for honnig-i with an impulse. COROLLARY VIII.1; There can be no minimum path against an outgoing target. One condition of homing is G <(P The condition for a target to be outgoing is 9 _ 90. These two conditions are incompatible with equation (8.1). We defined r" = 1 - e c and we saw (Corollary VI.3) that if the allowed unit fuel consumption r exceeded r_ we could eff.ect homing. THEOREM VIII.B: For a minimum path the unit fuel consumption is less than r". See analytic proof of Theorem V for details. Page 62

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 Proof. We can eliminate 9 between equations (7.3) and (7.9) to get qim = V sin Gm + U cos Gm = q cos ( - 9) or (8.3) qm < Q since ~ > e The unit fuel consumption is then qlmr rm = 1-e C Q < 1 - e whence (8.4) rm < r. W-e had a lower bound to unit fuel consurl:ption V r'= 1 - e c for incoming targets. This was the least unit fuel consumption that could possibly effect homing (equation (6.3) following Theorem VI). COROLLARY VIII. 2. For a minimum path the unit fuel consumption rm is greater than or equal to r'. If thrust is iinite, then r, > r' Page. 6 Pa~~~~e ~~~~S~

AERONAUTIC'AL RESEARCH CENTER UNIVERSITY OF MICHIGAN.......~r Report No UMM 18 Proof. Consider equation (8.3) qlm = Q cOS ( - 9m) > Q, cos ( 9 - 900) since a > 9m - 90o. We saw thet if thrust was finite the strict inequality held: (8.5) qzm > Q sin ( for acceleration bounded. Hence we have bounds for rm when thrust is finite. (8.6) rt < rm < r" - Pa;e 64

AERONAUTICAL RESEAARCH CENTER - UNIVERSITY, OF MICHIGAN Report No UMM 18 IX. SPECIAL MINIMUM PATHS In the previous sections, the work on minimum fuel consumption was done under the assumption that the second fundarental principle of homing with minimum fuel consumption (that thrust be large early and then cut off, see Theorem V) was observed. This included the case where thrust is a specified function of time fbr t < tt, and the case where throttling could take place iith either thrust or acceleration having a known maximum value as a function of time. We consider now a special class of paths of minimum fuel consumption. Wle consider two particular cases, then a general class of problems including both of them. A) Consider the following problem: let it be decided to home with acceleration constant all durin, homing, the constant being properly chosen to effect horing. The first principle of homing, that thrust must be fixed in direction, must be satisfied. Hence, we can write the position of the craft -~ ~ - Page 65 -

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 x = 1/2 a(G)t2 cos 9, (9.1) y = 1/2 a ()t2 sin, and the condition of homing 1/2 a t2 cos - Ut2 - = (9.2) 1/2 a tz2 sin 9 - Vt = 0. The integral (9.3) 1 a(9) dt = t2 a(g)' 0, is to be minimized. We write a(9) to emphasize that the value of the required acceleration depends upon the angle of thrust. (9.4) t2 a(9) - 2V sin 9 and now we wish to minimize the left side. Since V is known, the right side attains its minimum for (95) 9 = 90; that is, for minimum fuel consumption, the thrust must be applied along the vertical to the initial line of sight. Page 66

AERONAAUTICAAL RRESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 B) Consider the following case: let thrust T be constant during homing, the constant chosen to effect homing. Using the equations of Section II, we can write the condition of homing ct2 [ + (1- 1) ln (1- r) cos 9 = Xo+ Ut2, (9.5) ct2 [1 + (1 - 1) In (1 - r) sin Q = Vt. We can rewrite the second of these as (9.6) L[ + (~- 1) in (1 - r)] sin - r c sin 0 =constant sin 9 Now the function 1+(~1 ~r ~ r 1 + (- 1) ln (1 - r) = 2 6 + 1 - + n(n+ 1)+ * is clearly monotonic increasing in r. Hence to minimize r we. need to minimize the right side of equation (9.6). As before, it attains its minimum for = 900. C) General Case: these are all examples of the following case. Page 67

AERONAUTICAL RE.SEARCH CENTER - UNIVERSITY OF MICHIGAN --... Report No UMM 18 THEOREM IX. Whenever the velocity required for homing can be expressed in the form (9.7) v = f(A,t/t2), where f is an increasing function of A for any fixed value of t/t2, minimum fuel consumption is achieved for thrust applied at 90~ from the initial line of sight. We consider only incoming targets; a;ainst outgoing targets there is no minimum. Proof: From the results of section IV we must minimize (9.8) v = f(A,) a monotonic function of A; that is we need to select the angle of thrust 9 to minimize A. For homing to occur we must have (9.9) S2 = S2 Now, Vt2 (9.10) S2 sin and s(A,t2) = v dt Page 68

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 jt2 s(A,t2) = f(A, t/t2) dt J0 tj f(AT) dr or (9.11) s(A,t2) = t2 F(A), defining F(A), a non-decreasing function of A. Hence, by relations (9.9), (9.10) and (9.11) it is required for homing that V F(A) = sin 9 The right side of this equation has its minimum when 9 = 900. Hence A attains its minimum when 9 = 900, v2 has its smallest value for 9 = 900, and minimum fuel consumption is attained for 9 = 900. Corollary IX.1. The conditions.(2) can be replaced by the condition thct v2 = f (A,l) be a monotonic increasing function of F(A) = f(A,r) dr J0 This was the only property of f used in the proof. - - Page 69

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No,UIVM 18 Corollary IX.2. If the acceleration has the form a = g(A,t/t2) where g is an increasing function of A, thrust must be applied at 90 from the initial line of sight to achieve minimum fuel consumption. If the acceleration has the form a g(A,t/t), t 0 v - t g(A,t/t. ) dt rt/t1 = - g(A,) d'r, This we can rewrite as v = f(A,t/t2) and this velocity function satisfies the hypotheses of the theorem. Corresponding to this first corollary we could specify Weaker conditions on the acceleration function. It is required only that J g(A,r) dT be a monotonic increasing function of fl (1 - or) g(A,r) d'r. Page 70

AERONAUTICAL RE-SE;ARCH CENTER UNIVERSITY OF MICHIGAN Report No UMM,18 In paragraph A of this section we considered the case when the acceleration a was constant during homing. We wrote this as a(G) to indicate'that the value of this constant depended upon the direction in which.thrust was applied. We could equally well have written a(t2) since e and t2 are connected by the formula tan 9 = Vt2 Xo + Ut2 If we set A(t2) = t2 a(t2) we see that v = A t/t [= f(A,t/t2)] v2 = A [- f(A,l)] s2 = t2A [= t2 F(A)] so that f(A,1) is a strictly increasing function of F(A), the hypothesis for corollary IX.1. This, the example of constant acceleration, is the second most simple example of the theorem, leading to the case when v is linear in A and linear in t/t2. The case of the impulse is the simplest; v is linear in A and constant in t/t. l Corollary IX'.3. If the unit burning rate r has the form | r= h(A,t/t2) with h an increasing function of A the conelusion of the theorem holds. |^~~ 9 ~~~Page -71 ~,E|

AERONAUTICAL RESEARCH CENTER UNIVERSITY OF MICHIGAN Report No UMM 18 We see that r = r dt - 1 h(A,t/t2) dt /t h(A,r) dr = H(A,t/t2) a strictly increasing function of A. Now v = - c n ( - r) = - c In l - H(A,t/t2)] Since an increasing function of an increasing function is an increasing function, the hypotheses of the theorem are satisfied. In paragraph B we considered the case when thrust was constant during homing, this constant being properly chosen to effect homing. In any given situation r = r(t) since the value depends only upon the homing time chosen. -- —;- Page 72

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UIM 18 In this case r(t,t2) = r(t) dt = t r(t); this we can rewrite r(t,t2) = t2 ((t2)' t/t2 Now, if we set t2 r(t2) = A, this assumes the form of the function for fuel consumption in corollary IX.3, r = A t/t [= f(A,t/t)] This is the case when fuel consumption can be expressed in a particularly simple form, linear in each variable A and t/t2. Velocity then has the form v = - c In (1 - At/t) ~Page 73 - ~

AERONAUTICAL RESEARCH CENTER UNIVERSITY OF MICHIGAN.... ~Report No UMM 18 APPENDIX 1 DISCUSSION OF COORDINATE SYSTEMS The first problem in a study is to choose a proper coordinate set in which to work. We shall see that any cartesian coordinate set whose axes are not rotating in space may be used equally well. We shall see that the fundamental equations depend only upon the initial relative position and the initial relative velocity of the target with respect to the craft and upon the thrust applied to the craft. They are independent of the actual velocity of either the craft or the target, of body forces, and of arbitrary displacements of the origin of coordinates. This allows a great deal of freedom in the choice of a coordinate system. Since Newton's laws furnish the basis for our study, we shall start with a Newtonian reference frame or coordinate 1The material of this section is essentially the same as in many books on mechanics and theoretical physics. See, for example, Synge and Griffith, Principles of Mechanics, Chapter XII, or G. Joos, Theoretical Physics. Page 74

AERONAUTICAL RESEARCH CENTER UNIVERSITY OF MICHIGAN Report No UMM 18 system. Let x be the position vector of the craft and X the position vector of the target in this reference frame. The equation of motion of the craft is then 2M d x T MG dt2 where MG is the body force. We can rewrite this as (A.1) d2 = cri +Gl dt2 1 - r as in Section II; r is the unit vector parallel to T. Let - c r T a = 1 - r then a is the vector of acceleration due to thrust. The integration of equation (A.1) leads to t Ft d: (A.2) t () + a dt + G dt dt dt This can be integrated once more to yield (A.3) x = xo + t (d- ) d o rd+ G drTdT J0 0 0 0 Consider a new coordinate system, the position of the origin of which is given by the vector q(t). Then the position vector x' of the craft in this new system is Page 75

AERONAUTICAL RESEARCH CENTER-UNIVERSITY OF MICHIGAN Report No UMM 18 (A.4) = q- j{ d d f -f = o + t() dr + dd- (t) O JO J0 0 If X' denotes the position vector of the target in this new system, then we have (t - (A.5) X = X + t(d)o + G do- dr - q(t) The coordinates of interest are those that give the position of the target with respect to the craft. We see that (A.6) Xt - X - x dX dr - t(dt dt)o- The only terns that enter in these equations are those giving the relative position at the time t = 0, the relative velocity at that time, and the thrust applied to the craft. Hence for the two-body problem we can relax the restriction that the system be Newtonian, so long as the axes are not rotating. The importance of this is that the origin can be moving. For example, a point on the earth's surface -Pay be taken as origin without influencing the fundamental equations. 1Account must be taken of the earth's rotation. Page 76

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 It will be helpful later if we choose q in the following manner t Ir q = x0 + t(d)o + Gf d- d. In this system t T (A.7) x = a da d d, dX d3E (A.8) X- -o (A.8) X' = (X - x)o + t(dt dt) 0 dt We see that x' depends only upon the applied thrust and X' depends only upon the initial relative motion. Let us consider a rectangular coordinate system in which x, i, z are the coordinates of the craft position, and X, Y, Z are those of the target. Equation (A.7) is equivalent to the three scalar equations t T x = I ax(r) dcr dr t rT (A.9) y = ay( ) d- dr rt rIT z = az() dr rdr where a = is the x-component of the acceleration due to a M -- thrust, etc. Page.77 ~

AERONAUTIOCAL RESE.ARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 Equation (A.8) can also be expressed X = Xo +Ut (A.10) Y = Y + Vt Z = Z + t. If we let ~ = X-x (A.11 ) l = Y - y f = Z-z expression (A.6) has the form -t Fr < = Xo + Ut - ax dr d J0 J0 ntr" (A.12) = Y + Vt - -ay d d J0 J0 t or = Zo + Wt- a dr- dr. Jo Jo In general we shall choose the orientation of our axes so that Yo = Z= W = 0.. This can always be in the following manner. Set up the square array1 representing a rotation of coordinates 1For example, see Snyder and Sisam, Analytic Geometry of Space, Article 37. Page 78

AERONAUTIOAL RESEARCH CENTER- UNVERSITY OF MICHIGAN Report No UMM 18 x y z XI.11 212 13 (A.13) Y' 221 R22 23 z' R 31 232 A33 where the 2's represent the cosine of the angle between the various axes; -12, for example, being the cosine of the angle between the old y and the new x' axes. Then the transformation is represented by x' - Q11 x + R12 Y + 13 z, etc. The quantities Xo, Yo, Zo, U, V, W, being components of vectors, are transformed in a similar manner. If we choose 3i, R32, 33, to satisfy the relations Zo' = 31 X0 + A 32 Yo + 33 Zo = 0 (A.14) W' - = 31 U + R32 V +R33W = 0 and 21, a 22, R 23, to satisfy the relation (A.15) Yo = R 21 Xo + 22 Yo + 23 Z = 0; then these conditions with the conditions of orthogonality and normality Page 79

AERONAUTIOAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 3 3 (A.16)..... l (A.G) 1= ij ik =l ji ki 1 if j = k 0 if j = k (j,k - 1,2,3) which the cosines satisfy, will determine the's except for three signs. We shall choose XQ to be positive, Vt to be positive and the orientation of the z-axis to form a right hand act of coordinates. Then equations (A.10) are in the desired form, we can drop the primes, X = Xo + Ut (A.17) Y = Vt Z = 0. Since ax, y, a are components of a vector, equations (A.9) for the rocket motion are unchanged in form by the rotation. Equations (A.12) now have the form 4 = Xo + Ut - ax dr d 4' = | (A.-18) a\ = Vt ay d- d 0t r0~ = =-i a dd - d dr Jo Jo ~'~ ~ ~...... Page 80

AERONAUTIOAL RE-SEARCH CENTER -UNIVERSITY OF MICHIGAN Report No UMM 18 It follows from the nature of vectors that it is not essential whether the subtractions (A.6) or the rotations are,performed first. The integral Vax + ay + az dt in the fundamental equation (2.5) connecting fuel consumption to thrust is the integral of the magnitude of the vector a = T, hence is not affected by a rotation of coordinates. M The condition that interception occur without further thrust is that the vector of relative position (Xo, Yo, Z) is parallel to the vector of relative velocity (I, V, W1) and opposite in sense, that (A.19) X = Y = Zo (= - t) - - t. U V W This can be seen by substitution into equation (A.12). Terms in. expression (A.19) with numerator and denominator both zero mtust be omitted. A relation corresponding to equation (A.19) must be satisfied at the end of burning if homing is to be achieved without further corrections. This problem is discussed in Section III. We have just seen that we can express the problem as a plane one. If we solve the plane problem, we can solve Page 81

AERONAUTIOAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Report No UMM 18 the problem in space, taking into account the forces of gravity, etc,. The real importance of the transformation is this: to solve the problem of homing it is necessary to go to the plane problem; this may be done explicitly, or it may be done implicitly. It is not necessary to go to the trouble of carrying out the above computations but only to determine Xo, U and V1. A second significance of this transformation is this: we can study the motion in the plane above and know that- we have not omitted anything of significance to the space problem, subject to the assumptions in the introduction. This is done in UMM-19, Section VI. ^*- ~ -~ ~~~~ Page 82.

AIERONAUTICAL RE-SE.ARCH CENTER-UNIVERSITY OF MICHIGAN Report No UMM 18 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. Page 83

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