GE? SWGE RENDEZVOUS WNEMUAR ENER Thegle; €02? the E‘MCWGAR STATE. U REVERSE'TY {Zagreza a? 2%; fig Eugem Harrigm 11962; 0) This is to certify that the thesis entitled Minimum Energy Space Rendezvous presented by Eugene Harrison has been accepted towards fulfillment of the requirements for Ph.D. degree in Mechanical Engineering WA’IZe Major professor Date July 3, 1962 0-169 LIBRARY Michigan State . Universit 3 Y I” ABSTRACT MINIMUM ENERGY SPACE RENDEZVOUS by Eugene Harrison The space rendezvous maneuver is defined as one which is designed to move a space vehicle from one location to another in order to match the position and velocity of an object in orbit at the second loca- tion. It is equivalent to a space transfer in Which terminal conditions and time of transfer are specified. An investigation to determine the minimum enerav re- quired to achieve a space rendezvous is reported. Two methods were used to investigate the rendezvous enerev problem-—an analysis bv the calculus of vari- ations, and a trajectorv perturbation procedure. For each method it is assumed that the onlv forces acting on the rendezvous vehicle are due to the inverse square gravity field and applied thrusts. It is also assumed that thrust levels are hiah so that velocity changes are essentially impulsive. The analysis by the calculus of variations is based upon the use of linearized equations of relative motion which are reasonably accurate if the distance separating vehicle and target is in the order of 50 miles or less. Using the relative equations, a set of Euler-Lagrange equations were obtained; which, on examination, led to Eugene Harrison the follOWing conclusions: (1) A minimum energy trajectory is comprised of seg- ments flown with either zero or maximum thrust. (2) A minimum energy rendezvous is accomplished using either two or three impulses. The perturbation method of energy analysis deter- mines the effect of perturbing a vehicle which would otherwise move along a nominal coasting trajectory join- ing space terminals. The perturbing velocity is assumed to be the residual from a partial nullification of the initial velocity relative to the coasting trajectory velocity. At an intermediate time an impulse is applied to bring the vehicle back onto the nominal trajectory at the destination point. The trajectory from the point of application of the intermediate impulse to the desti- nation point is selected so that the total transit time on the perturbed path equals the transfer time along the nominal trajectory. The velocity at arrival via the two routes would, of course, be different, and the vector difference is termed the resultant velocity. A third impulse potential is defined in terms of the residual, intermediate, and resultant velocities. The results of a parametric study in which the third impulse potential was used to examine the conditions under which a third impulse could be utilized are reported. A sample problem is presented in which the velocity required to rendezvous is shown to be greatly reduced by using three velocity impulses instead of two. MINIMUM ENERGY SPACE RENDEZVOUS BY Eugene Harrison A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1962 ACKNOWLEDGEMENT The author is indebted to several persons, both at Michigan State University and Chance Vought Corporation, for their assistance and encouragement during the conduct of the rendezvous investigation and the preparation of this thesis. Special appreci- ation is extended to Dr. F. S. Tse, the author's major professor, for his guidance and suggestions. Appreciation is also extended to Dr. R. T. Hinkle of Michigan State University and Messrs. F. T. Gardner, W. C. Schoolfield, and A. I. Sibila of Chance Vought Corporation for their assistance and constant encour- agement. To my wife, Melba, I give special thanks, not only for assistance in typing this thesis, but for bearing the major share of the responsibility for raising our family during the conduct of the thesis WOrko ii TABLE OF CONTENTS Chapter Page I. INTRODUCTION. . . . . . . . . . . . . . . . 1 II. PROBLEM DEEINITION AND BACKGROUND . . . . . 3 PROBLEM STATEMENT. . . . . . . . . . . . RELATED STUDIES 0 O O O O O O O O O ‘ O O Rendezvous Equivalent Transfer. . . . Orbit Transfer. . . . . . . . . . . . ANALYSIS METHODS USED IN SUBJECT STUDY . co m o: 4: .p 3 GENERAL ASSUMPTIONS. .-. . . . . . . . . III. ANALYSIS BY THE CALCULUS OF VARIATIONS. . . 11 IV. ENERGY ANALYSIS BY TRAJECTORY PERTURBATION. 18 PERTURBATION METHOD. . . . . . . . . . . 18 Description. . . . .. . . . . . . . . 18 Third Impulse Potential . . . . . . . 20 Determination of Nominal Coasting TraieCtorlo o o o o o o o o o o o 22 Calculation 2£_Third Impulse POtentTalo o o o o o o o o o o o 24 PARAMETRIC STUDY . . . . . . . . . . . 27 Secondary Effect Parameters . . . . 28 Primagy Effect Parameters. . . . . 32 Sample Problem. . . . . . . . . . . 46 iii Chapter Page V. SW'IMARY AND CONCLUSIONS. 0 o o o o o o o o o 51 CALCULUS QEVARIATIONS. . . . . . . . . . 51 TRAJECTORY PERTURBATION o o o o' o o o o o 52 VI. SUGGESTIONS FOR FUTURE STUDIES . . . . . . . 54 EX‘I‘ENSIONQEDATA..... 54 PRACTICAL COMPUTATIONS . . . . . . . . . 54 BIBLIOGRAPI-IYO O O O O O O O 0 O O O O O O O O O O 58 Appendices A. EQUATIONS OF MOTION AND ORBITAL MECHANICS. 61 ORBIT-Ml MOTION. O O O O O O O O O O C O 61 EQUATIONS OE RELATIVE MOTION. . . . . . 64 B. NOMINAL TRAJECTORY EQUATIONS. . . . . . . 69 C. TRAJECTORY PERTURBATIONS USING RELATIVE EQUATIONS. 0 O O O O O O O O O O O O O 74 iv LIST OF TABLES Table Page I. RESULTS OBTAINED WITH VARIOUS RADIUS RATIOS AND RESIDUAL VELOCITIES. . . . . 29 II. TRANSFER ELLIPSE ELEMENTS CORRESPONDING TO DIFFERENT TRANSFER TIMES. . . . . . 33 III. RESULTS OBTAINED WITH VARIOUS TRANSFER TIES. O O O O O O C O O O O O O O O O 34 LIST OF FIGURES Figure Page 2.1 ORBITAL TRANSFER WITH TIME UNSPECIFIED. . . . 7 2.2 WAITING TIME DESIGNED TO CORRECT PHASING PRIOR TO RENDEZVOUS. . . . . . . . . . . . 7 4.1 AXIS SYSTEM ON NOMINAL COASTING TRAJECTORY. . 19 4.2 RELATIVE TRAJECTORY OF PERTURBED VEHICLE. . . 19 4.3 INERTIAL TRAJECTORIES AND VELOCITIES. . . . . 26 4.4 ILLUSTRATING THE REVERSE EQUIVALENCE OF A TRAJECTORY. . C O O C O O O O 0 C O O O C 11 4.5 THIRD IMPULSE POTENTIAL. . . . . . . . . . . 56 4.6 INTERMEDIATE IMPULSE REQUIRED TO RETURN VEHICLE TO NOMINAL TRAJECTORY. . . . . . . 57 4.7 RESULTANT VELOCITY AT END OF TRANSFER . . . . 58 4.8 DIRECTION OF RESULTANT VELOCITY AT END OF TRANSFER. . . . . . . . . . . . . . . . 59 4.9 THIRD IMPULSE PARAMETERS . . . . . . . . . . 40 4.10 THIRD IMPULSE PARAMETERS . . . . . . . . . . 41 4.11 THIRD IMPULSE PARAMETERS . . . . . . . . . . 42 4.12 THIRD IMPULSE PARAMETERS . . . . . . . . . . 43 4.13 THIRD IMPULSE PARAMETERS . . . . . . . . . . 44 4.14 CURVES OF THIRD IMPULSE POTENTIAL PEAK VALUE. 45 - 4.15 COMPARISON OF TRAJECTORIES FOR LARGE ANGLE TRANst O O O O C C O O O O O O O O O O O _O 4'7 4.16 THREE IMPULSE CHARACTERISTIC VELOCITY CURVES FOR SAMPLE PROBLEM. . . . . . . . . 50 6.1 CURVES COMPARING EXACT TO EMPIRICAL EQUATION VALUES. 0 O O O O O O C O O O O O 56 A.1 ELLIPTICAL ORBIT GEOMETRY.'. . . . . . . . . 62 A.2 RECTANGULAR COORDINATE SYSTEM. . . . . . . . 65 vi LIST OF SYMBOLS PR IMARY SYMBOL S A 2 Aql . Aq ol .AE Matrix or constant as noted at usage Acceleration components in the generalized directions q1 and q2 Radial and tangential components of thrust- ing acceleration Thrusting acceleration components Apogee of transfer ellipse Dimensionless group Matrix coefficients A constant Dimensionless group Constant or Erdmann-Weierstrass corner con- dition as noted at usage Rocket exhaust velocity A constant Eccentric anomaly Difference between the eccentric anomalies of two points on an orbit vii en 81 <| 44V 4| Eccentricity of transfer ellipse Gravity force per unit mass Function of relative equation parameters Angular momentum Mass of rocket and fuel Average angular velocity of an object in orbit Period of orbit or space terminal as noted at usage Generalized parameter defined at usage Radius of orbiting object measured from the earth's center Total transfer time Time Relative velocity in x direction Absolute or inertial velocitv Characteristic velocity Relative velocity (in y direction or total magnitude as noted at usage) Relative velocity vector viii 5v 19Y:x Y1 5Y1 Ii 51 .62 Third impulse potential Rectangular coordinates Relative position and velocity parameters in first order equations of motion Variations from trial trajectory Adjoint equation parameters Angle between the thrust and velocity vectors -a Direction of residual velocity at initial space terminal Direction of resultant velocity at destination Flight path angle measured between velocity vector and horizontal direction Phase angle Ratio of the areas of the triangle and ellipti- cal sector between two radii of an orbit Lagrange multipliers Earth gravity constant . 1.4077998 x 1016 ft3 see Time dependent variable used to limit the thrust magnitude ix A?) Radius from earth's center to vehicle Time measured from point of intermediate thrust application True anomaly of transfer ellipse Constraint equations Difference between the true anomalies of two points on an ellipse “J Angular velocity of circular orbit (°) Denotes differentation with respect to time a: ( ) Denotes evaluation along a trial trajectory SUBSCRIPTS cl,c2 Initial and final values on a nominal coasting trajectory f Final value 1 Intermediate value i,k,q Identification subletters which take successive values as noted at usage n Nominal 1,2 Denotes starting point and destination on a transfer trajectory CHAPTER I INTRODUCTION The purpose of the investigation described in this thesis was to determine the minimum energy required to transfer a spacecraft between two points in an inverse square gravity field when the time to transfer and terminal velocities are specified. With these specifi- cations, the transfer is equivalent to ”space rendezvous" -a maneuver generally described as one designed to move a spaceCraft from one location to another in order to match the position and velocity of some object in orbit at the second location. Hereafter, the designations "vehicle" and ”target" will be used. The need for information concerning the rendezvous problem is becoming increasingly important. A number of complex satellites have already been placed into orbit and many more are planned for the near future. Some of these will be manned and may possibly range from craft carrying a single operator to facilities that will serve as launching platforms for lunar or interplanetary mis-. sions. Rendezvous capability will be necessary in order to transfer personnel and supplies to and from these facilities and to provide emergency rescue capability. Recent literature has been replete with articles concerning rendezvous. One author lists a bibliography containing fifty-nine references.1 For the most part, these articles have dealt with the development of guidance 1 equations and various rendezvous techniques. The energies required to rendezvous by the various techniques differ, and in many cases by a significant amount. One of the reasons for attempting to determine the minimum energy required for rendezvous was to provide a comparison energy so that an efficiency number, insofar as energy or fuel expenditure is concerned, may be determined for each technique. The total operation from earth launch to contact between vehicle and target may be described as consist- ing of the following phases: (1) Launch from earth for a direct approach or into a parking orbit. (2) Orbit transfer and/or injection into the target orbit to get an approximate position and velocity match. (3) Rendezvous, which is usually described as be- ginning between 10 to SO miles from the target and extending to close proximity. (4) Final approach and docking which begins within a few feet of the target. It was intended that the subject investigation would be restricted to the rendezvous phase, however, in many cases the results obtained could equally well apply to the orbit transfer phase. CHAPTER II PROBLEM DEFINITION AND BACKGROUND PROBLEMiSTATEMENT In terms of the generalized coordinates q1 and q2, the subject problem can be stated as that of transferr- ing a spacecraft or rocket from condition (q},q§,q%,q3) to (q:,q§,q:,§:) in time T with a minimum expenditure of energy. The motion of a rocket in space can be des- cribed by the vector equation d5 +F= -(C/m) 93 A (2.1) d? dt where ‘F = rocket velocity mass of rocket and fuel 8 I = rocket exhaust velocity *sl 0| - gravity force per unit mass. Energy expended is usually measured in terms of char- acteristic velocity as determined by the integral of either side of Eq. (2.1). Hence, the energy of transfer (rendezvous) is determined either by the equation "Wt AV '- /-('é'/m) gm =- c log (mo/mp) (2.2) dt mo or v1. 7 AV=/dF+/th. (2.3) V0 0 The problem of minimizing energy can be considered as one of minimizing the change in mass, (m0 - mT), or the characteristic velocity as defined by Eq. (2.3). RELATED STUDIES Previous studies pertaining to the transfer of a vehicle between space terminals more or less fell into two general categories: (1) those in which time is specified and are rendezvous equivalent, and (2) those in which time is not specified. A number of the more pertinent investigations and their relationship to the subject investigation are briefly discussed in the fol- lowing paragraphs. Rendezvous Equivalent Transfer In early investigations Lawden used the classical variational calculus approach to analyze the space trans- fer problem.3’4’5 On the basis of a Taylor series expan- sion of the gravity potential in which terms greater than second order were neglected, results were derived which indicated that the minimum energy transfer trajectory 'would contain no segment flown with an intermediate thrust level, 1.6. it would consist of arcs of maximum thrust followed by coasting arcs. However, both Lawden and Leitmann later showed that an intermediate thrust trajec- tory cannot be ruled out when the complete gravity poten- tial is present.“7 In reference (7) Leitmann treats 'the problem of thrust mode selection and shows it to depend upon a switching function determined by the Weierstrass E-function. However, this function does not furnish apriori information for the above stated case (a non-linear gravity function). A running account must be kept of the switching function during the numerical integration of the Euler-Lagrange equations. Thus, one of the primary problems in determining an optimal trajectory is the selection of the appropriate thrust mode. However, even going under the premise that a minimum energy trajectory in an inverse square gra- vity field would contain no intermediate thrust segment, there still remains a major problem. This problem is the lack of criteria for determining the number of powered arcs to achieve an optimum. In many cases where the classical calculus of vari- ations was used to analyze those Space transfer problems which lend themselves to analysis, the procedure has been to formulate criteria for solution without present- ing a solution. The difficulty lies in solving the re- sulting set of Euler-Lagrange equations. It is necessary to know all of the initial conditions before the equations can be numerically integrated. In most cases, terminal conditions are specified in part as initial, and in part as final conditions. Thus, an iterative scheme must be employed to obtain a solution. To avoid the difficulty associated with the use of the methods of the calculus of variations, a number of investigators have proposed the use of direct methods of optimization. Among those that have been employed for trajectory optimization are the gradient theory methods, 9,10,11 and a method analogous to the Rayleigh-Ritz methods. Reference (12) is a survey of the problem of opti- mizing aircraft and missile flight paths. Prior treat- ments are described and the problems of Bolza, Mayer, and Lagrange are developed. This reference contains an excellent bibliography pertaining to trajectory opti- mizing. ngit.Transfer The orbit transfer problems which specify terminal conditions but not the transfer time are not equivalent ‘ to the rendezvous problem, although some insight may be drawn from them. The difference may be seen in Fig. 2.1 which shows two coplanar orbits that are nearly identi- cal in geometry but differ in orientation. Let it be assumed that an optimal orbit transfer trajectory would be similar to path Pl‘P2: where one impulse is applied at P1 on orbit A to initiate the transfer trajectory and a second is applied at P2 to enter orbit B. Now, it wouki be coincidental if the vehicle and target were phased in orbit such that this trajectory would result in an intercept. Further, since the orbital periods would be nearly equal, waiting until the proper phasing occurred might not be feasible. However, if time is of no conse- quence, the fuel required to rendezvous would, in the limit, be equal to that required to transfer orbits. Such a rendezvous would be carried out following the orbit transfer by applying a small retro thrust at the perigee (for minimum velocity impulse required/period change) of the entered orbit in order to go into a wait- FIG. 2.1-ORBITAL TRANSFER WITH TIME UNSPECIFIED FIG. 2.2-—WAITING ORBIT DESIGNED TO CORRECT PHASING PRIOR TO RENDEZVOUS ing orbit with a slightly shorter period (Fig. 2.2). The waiting orbit should be tailored so that the vehicle and target would arrive at their coincidental perigee points simultaneously after a number of revolutions. One of the first works concerned with the minimiz- ing of orbit transfer energy was the derivation by Hohmann of the transfer ellipse named for him.13 The method uses two impulses to transfer between circular orbits. The first is applied at a point on the initial orbit to establish an elliptical trajectory tangent to the two orbits. Transfer is accomplished during 180° of orbit travel, and another impulse is applied to enter the second orbit at the point of trajectory tangency. Transfer can be to an orbit of either smaller or larger radius. The Hohmann transfer ellipse provides a minimum energy transfer if the radius ratio between the two orbits is not greater than 11.9.14 Other investigators have examined the more general problem of transfer between non-circular orbits. It has been shown that the optimum number of impulses to transfer from one coplanar orbit to another is either two or three, depending upon their orientation and ele- Iment values.15:16 ANALYSIS METHODS USED _I_1_\I_ SUBJECT STUDY Early in the program three avenues of investigation were followed, more or less simultaneously. The three consisted of a study of the applicability of the calcu- lus of variations and two direct minimization methods. 9 One of the direct methods, a traiectorv perturbation technique, was the primary tool of subsequent investi- gation. The other direct method of analvsis was based upon a variation of delta quantities to establish a gradient that would indicate the manner in which a non- minimum trajectory should be varied to reduce transfer energy. It was abandoned because of excessive computer time requirements and lack of proof that the determined trajectory was an extremum and not merelv one with a stationary energy level. Treatment of the rendezvous energv problem by the methods of the calculus of variations and the traject- ory perturbation technique are presented in Chapters III and IV respectively. GENERAL ASSUMPTIONS The fOIIOWint assumptions were made in order to facilitate the energy analysis and are not believed to detract from the results obtained: (1) The earth is spherical. The perturbations of a satellite orbit caused by the earth's oblateness may be neglected insofar as rendezvous is concerned because the integrated effect of the perturbations would be small during the time interval of interest.2 Also, since the distance between target and vehicle during rendezvous is small compared to the radius of the earth, the perturbation effects would be approximatelv the same on each and (2) (3) (4) (5) 10 would cancel insofar as relative motion is concerned. , . Gravity is the only outside disturbing force. The forces exerted by atmospheric drag, meteor- ite collision, and sun pressure are neglected. The target and vehicle are in coplanar orbits. Motion in a direction normal to the target orbit plane is, for all practical purposes, independent of in-plane motion for the relatively small separation distances being considered. (See Eq. A.24.) Thus, it is possible to treat the in-plane and out-of—plane motion independ- ently and superimpose results. The vehicle is a point mass. The problem of attitude stablization and thrust vectoring was not investigated. Impulsive thrusts and instantaneous velocity charges are allowed. The time required for a vehicle to attain a desirable closing velocity in the usual situation would be small compared to the time to perform the terminal phase rendezvous maneuver. CHAPTER III ANALYSIS BY THE CALCULUS OF VARIATIONS The analysis by the calculus of variations pre- sented here relies heavily upon work described in references (5,7,12, and 17).‘ The problem of energy minimization is analyzed using the linearized ren- dezvous equations (A.22 and A.23). Within the lim- itations of these equations, which are shown in Appendix A to be reasonably accurate, it will be shown that a minimum energy rendezvous trajectory contains no arc flown with intermediate thrust. In addition, it will be shown that if thrust is unbounded, a minimum energy trajectory is achieved with either two or three impulses. If the angle between the thrust and velocity vectors is denoted by q, Eqs. (A.22 and 11.23) can be written as the following set of first order equations: 40! ' i ' u = 0 (3.1) (Py'i-v-O (75.2) (on " {I - 2M7 - (ed/m) Cosot= O (5.3) $7 " 9 + 2141 - Sway - (ob/m) Sin ou- o (3-4) {om-fine- o . (2.5) Assuming the rocket exhaust velocity, c, to be a constant, as is nearly so in the case of a chemical rocket, there are two control variables—«and 6. ll 12 Following Lietmann,7 a sixth equation is introduced in order to limit the mass flow rate such that 3min 5 5 S‘Gmax . The equation is w; . (/3- fiminHGmax -6) -42 = o ' (3.6) where g-ém . Equations (3.1-3.6) are six restricting conditions to be imposed on the eight variables x, y, u, v, m, Gt, 6, 4‘, leaving two degrees of freedom. The problem at hand is to determinelq(t) and 48(t) such that the rocket will traverse a trajectory between designated space terminals in time T that minimizes (mo - mT). As a first requisite for a so- lution, the Euler-Lagrange equations must be satisfied. These equations are g_f2§) -:2§_ - 0; q . x,y,u,v,myay6y€ dt49§ 8n_ . where F -XM¢Q; q = X:Y:u:v9m:€ and the kq are the Lagrange multipliers. When applied to Eqs. (5.1-3.6), the resulting set of equations are as follows: 13 xx = o (3.7) iy + 352w = o (3.8) in - away + Xx - O (3.9) iv + swap + Ky - O (3.10) in - (ca/mszu Cosq + Xv Sinq) = O (3.11) (c/m)(?»u Cos OL+ 7w Sin 04) - M (3.12) + 7mg“)? namin) - (enlax -/3)] = O (ca/m)(7\.u Sin or - Xv Cos at) -= O (25.13) xgg- o (3.14) Solutions for Xx, Ky, Nu, and AN become immediately available by first differentiating Eqs. (3.9 and 3.10) and substituting from Eqs. (3.7 and 3.8), where- upon, the following equations are obtained: in - 201v . o (3.15) 33v + 2&9111 " SIDZKV a 0 e (3016) It is seen that with a substitution of Kn for x and Xv for y, these equations become identical to the linearized equations of motion (A.22 and A.23) with Ax - Ay - O. Leitmann proved in the case of an analogous problem, i.e. when the ku-Kv equations can be uncoupled from the remaining set of Euler-Lagrange equations, that no arc flown with intermediate thrust can exist.7 Thus, an optimum trajectory would be composed of arcs of maximum and minimum thrust. In the present investigation ve- locity impulses have been assumed, i.e. fimax +00. l4 Thrust direction is determined from Eq. (3.13) as TanOC- Xv . (7.17) XE’ ' It is noted that the functions Xu and Xv remain defined between impulses since 6min is allowed to approach but not attain a zero value. If an optimal trajectory is to be composed of more than one coasting arc, the Erdmann-Weierstrass corner conditions must be satisfied at each corner (interior junction). These conditions are expressed Gilli-E). (H) which define the equality of quantities immediately as prior to and after a corner. Applied to the present problem they result in M- = M. 3 q = X.Y.u,V.m (3.18) c- = 0,, where C = Xxi +.ny + qu-+ va . First,it is noted that Xu and Xv satisfy Eq. (3.18) by the noted solutions (analogous to x and y which are continuous on a coasting trajectory). Next, parameters 15 Xx and Xy satisfy this corner condition because Xx is a constant by Eq. (3.7), and from Eq. (3.8) Xy = -3u§‘/{Xv dt (3.20) and so is also continuous. In order for Eq. (3.14) to hold, it is necessary that X; = 0, since €(t) 9‘ 0 by definition. Hence, from Eq. (3.12) M *3 (c/m)(Xu Cos c14- Xv Sin a) (3.21) and, since c = 0 immediately before and after an impulse, Xm..- Xmun Thus, all of the corner conditions of Eq. (3.18) are shown to be satisfied. The corner condition specified by Eq. (3.19) furnishes one of a set of equations which can be solved to determine a stationary three impulse rendezvous tra- jectory. In total, twenty-seven equations can be written relating thirty-six parameters. From.Eqs. (3.1-3.4) with zero thrust and from Eqs. (3.15 and 3.16), explicit solutions can be determined for the following twenty parameters: A x1, 3T Xui, XuT Y1: VT Xvi, XvT ii—’ iT— Xui Ii-o IT— Xvi ‘31-: {114- M1 61” {71+ ~ Xyi . (Six equations remain to be defined. 16 At each application of impulsive thrust the direction of the thrust vector must coincide with the added velocity vector. Hence, Eq. (3.17) applied at each impulse results in the three equations Mk = 41x = YR: - 315.: k = 0.1.T - (3.22) Kuk Mk Xx. - X1:- ' The three other equations, which have been shown to be necessary conditions for an optimal trajectory,5 are [(Kuk)z + (K”k)2]1/2 ' l; k = O,i,T . (3.23) These equations state that Xu and Xv are not merely proportionsl to the thrust components, but in fact, are the direction cosines of the thrust vector. The desired unknowns of impulse direction, mag- nitude, and timing can be determined as functions of the eight terminal conditions and total transfer time by solving the set of twenty-seven equations described above. In order to see that no more than three impulses can be utilized for optimum transfer it will be re- called that Eqs. (3.19 and 3.22) must be satisfied at points of impulse application by the parameters Xu and Xv. Since these two parameters are described by second order differential equations between t = O and t = T, the four initial conditions Xuo, Xvo, Xuo, and Xvo are just sufficient to satisfy the conditions specified by these equations. Additional conditions that would be f‘ 17 introduced by another corner could not, in general, be satisfied. In summarizing the application of the calculus of variations to the problem of rendezvous fuel mini- mization, the following items are noted: (1) (2) Insofar as the linearized equations describe the relative motion of a rendezvous vehicle and target, it has been shown that-— (a) No arc flown with thrust between the maximum and minimum values can exist on a minimum energy trajectory. (b) The minimum energy transfer is via either two or three impulses. No criteria was developed to determine which of these two modes would provide the minimum. Since the linearized equations have been shown to be reasonably accurate, it would not be expected that there would be any sig- nificant difference in the above results as applied to the actual rendezvous maneuver. It should be called to attention that the linearized x-y equations are based upon an axis system whose origin moves in a circular orbit, but that the target need not. It is necessary only that the target position be predictable in the reference frame at time T. CHAPTER IV ENERGY ANALYSIS BY TRAJECTORY PERTURBATION In many respects this chapter presents the most important results obtained from the rendezvous energy study. The methods of analysis which were developed are described and a means for measuring the potential of a third impulse is formulated. The results of a parametric study to determine the conditions under which a potential exists are also presented. PERTURBATION METHOD Description The perturbation method, as aoolied to the rendez- vous energy study, determines the effect of perturbing a vehicle which would otherwise move along a nominal trajectory joining two space terminals. Let it be assumed that the origin of an axis system as shown in Fig. 4.1 moves along a nominal trajectory between the space terminals P1 and P2 as if it were attached to an undisturbed mass. Further, let the position and velocity of the origin be Pl"Vc1 at t = O, and P2, veg at t = T. Then, if the absolute velocities of a vehicle immediately prior to P1 and after P3 are denoted by?o and If, these velocities relative to the coasting reference frame are determined bv the vector differences v0 = Vb ‘ V51 18 19 A” _I.P1 FIG. 4.1-AXIS SYSTEM ON NOMINAL COASTING TRAJECTORY FIG. 4.2h-RELATIVE TRAJECTORY OF PERTURBED VEHICLE .MQ‘Q15 ‘) 1 ‘3 “I-“fim..-_- gag-.- -h -h.._...—,. I I I . 20 (Upper case letters are used to denote absolute velocities while relative velocities are denoted by lower case letters.) If impulses were applied according to the equations 'Vl ‘ ‘70 V2 " :61" the vehicle would.follow the coasting trajectory and remain at the origin of the reference frame. On the other hand, suppose that the relative velocity at P1 is not entirely cancelled, but that a residual velocity, v is allowed to remain. In 1. this case the mass would drift away from the frame origin. However, at an intermediate time, t = ti’ another velocity impulse,.4v1, could be applied that would bring the mass back to the origin at the moment P2 is reached. Relative trajectories similar to those shown in Fig. 4.2 would result. The energy expended would be the sum of the three impulses according to the equation Av = [‘61 - Fol +lAv1|+ IVr - Val. Third Impulse Potential In the case of a two-impulse rendezvous, the required characteristic velocity is determined by the equation AV - '57:” + [1.6521 .. lVol + (V3). Hence, if a three impulse transfer is to require less 21 energy than would be required by the use of only two, it is necessary that the energy denoted by Eq. (4.1) be less than that denoted by Eq. (4.2). Hence, it is necessary that I51 ..';;0| + (5‘71) + fir ' "72I < “'0' + lvrl (4.3) or (4‘71) < ”0' ' W1 " Vol "’ Ivf' " I“If " 17—0)- (4.4) Considering v0, v1, and (Vi - V0) as three sides of a triangle, it may be seen that (V1) > Iv.) - El -Vol (4.5) and likewise Ivzl > (Fr) " lVr - V2). (4.6) Upon substituting inequalities (4.5) and (4.6) into (4.4), it is found that lAvil < IVll "' (‘72). This expression is more convenient when written as the equality Xv = |v1| + |v2| - lav“, The quantity Sv was used as a criterion for deter- mining whether a third impulse is potentially useful for reducing rendezvous energy. A positive value sig- nifies that a potential exists. However, in order to realize the full potentia1,'v1 and?!2 must be parallel to'v'o and'vf respectively. At times, the 22 initial and final velocities may be such that the sum of the impulses required at P1 and P2 is greater than the potential corresponding to the selected magnitude and direction of v1, i.e. the quantity In this case a potential would exist that could not be realized due to the directions and magnitudes of the initial and final velocities. Determination 9; Nominal Coasting Trajectory The trajectory of an object coasting in space follows a conical path which can be either an ellipse, parabola, or hyperbola. However, the_subject invest- igation was restricted to a consideration of motion along an elliptical path. (Circular paths are included in this category.) The term "transfer ellipse" is used to designate the entire orbit of which the coast- ing transfer trajectory is a part. Thus, a transfer trajectory may be described by the ephemeris of the transfer ellipse. Unless some special orientation is assumed for the transfer ellipse, the equations of Appendix A alone are not sufficient to determine the ellipse elements if the known information consists of only R1, R2, up, and T. However, this information and the direction of orbit rotation does specify a unique trajectory. The orbit elements may be determined as follows:18 23 Let / .. __(g11 2T 81 [2(R1R2)1/2T Cos (Aw/2)]3fi? ' p1 = R1 + 3.2 - 1/2 4(R1R2)l/z Cos (AWZ) where E1 and E2 are the values of the eccentric anomaly at P1 and P2. Then, the parameter AE/2 can be deter- mined from one of the following equations which are derived in Appendix B: t al 3 [b1 + Sing (AE/4)]l/2 (40 9) - Sin AE) [b1 + Sin2 (AB/4H?) 32/ Sin:5 (AF/2) where the sign preceding a1 is taken as + for A¢< 180°, and - for A¢>1so°; and, for the case in which Ago= 180° (ml/3 th/(R + R 2)]3/3- - SinAE . (4.10) Sin3 (AB/2) These equations can be solved by trial and error, and with 4E known, other elements can be computed by the following equations: ZRle Sing (mo/2) (4.11) R1 + R2 - 2011123)“z Cos (Aw/2) Cos (An/2) a = 333 Sin2 (Aw/2) (4.12) pain2 (An/2) e - (a -p)1/2 (4.13) a 24 Terminal coasting velocities can be determined from equations (A.l4 and A.17). In subsequent references, the above described method of solution and the computer routine which was developed to solve the system of equations will be referred to as the “coasting routine". gglculationqgg_zhi§g.Impulse Potential After having determined a nominal coasting tra- jectory,"v1 and'yz can be computed for an arbitrarily selected value of'VI. Two methods of solution were developed. One makes use of the coasting routine described in the previous section plus an integration of the orbital equations of motion. In the second method, the exact relative equations of motion are integrated using an iterative procedure to determine the sought variables. The majority of computations were made using the coasting routine and orbital equation technique because it required considerably less computer time. For the average solution only about one-tenth as much time was required. However, the relative equation technique, which is described in Appendix C, was found to give highly accurate solutions and was used to check the accuracy of the other method. Following is a description of the coasting routine- orbital equation method for determining the quantities comprising 5v. After 52,1 and $02 are computed by the coasting routine, a selected value of Vi is added to 25 V01 to give a new absolute velocity at P1, which is shown in Fig. 4.3 along with the other vectors of inter- est. The next step in the solution is to determine the radial and angular velocities at P1, which can be computed from the equations 1321 = (Io). + '51) Sin r1 (4.14) 4’1 " E (Vol + 71) /R1] Cos r1 . (4.15) With the initial conditions at P1 known, the equations 1% - W2 + (q/Rz) = o (4.15) R<‘p‘ + add - o (4.17) are integrated to determine the position R1 and velocity [V1 at the intermediate time, ti. Following this step, the coasting routine is used once again to determine a coasting trajectory that will traverse the distance between P1 and P2 in the time remaining. Inputs to the routine this second time are denoted by primes and are Ri . Ri R; = R2 4?: OJ? - "pi - ([11] T a T - ti . The velocities V3 and V2 are determined at the ter- minals of the new coasting trajectory, and the sought parameters are given by the vector differences 26 P1 NOMINAL *\\\§&D ' _~P FIG. 4.3-INERTIAL TRAJECTORIES AND VELOCITIES 1!. ii 0 II 1.1!...1!‘ Ill ( )5 II). .i’tx‘Iéuiv l0 5!!” . .p a z r , ../ . w . I , < /\ / / r '3 V “I...“ m~mv-m / I ~ ~ / //I / .o / / _. I r ‘l I I! I. I: 1 A. . . l 0! .\\ 1-.-)? .ln . n no! .I .0.)th " 0|, . ()I nut. PO! 1.. .-l 1 til .15 I o... ,3 pi .r: . u .. , tun-up... [I \u \n f 27 vi 3 ‘51+ "fiflrl v2 ' ($2 -ECZI ' The total computer time required to determine a single value of 8v, including use of the coasting routine for arcs P1-P2 and Pi-Pz, and integration over are Pl‘Pi. was slightly less than two seconds. The accuracy obtained was limited by round-off errors in the computer (an eight digit retention routine was used). Ellipse elements p and a were determined with errors less than 1 ft; and, although round-off velocity errors as large as 3 ft/sec were possible, the usual noted error was in the order of 0.2 to 0.5 ft/sec. PARAMETRIC §E§2§ The perturbation procedure described in the previous section was used to make a parametric study of the third-impulse potential, 8v; The following parameters were varied during the study: Rg/Rl 8 ratio of final to initial radius T ' transit time A¢ 8 angle between R1 and R2 t1 = time of application of intermediate impulse v1 = magnitude of the residual velocity relative to coasting frame at P1 61 - direction of v1 (Fig. 4.2). It was found that the parameters could be divided into two groups-those having primary effect and those 28 evidencing only slight or secondary effect. Secondary Effect Parameters Small variations of R2/R1 and T from nominal values were found to have only a slight effect upon the output parameters comprising 5v, i.e..ayi, v2 and its directionqaé. In addition, it was found that the ratios awi/vl and vz/vl are independent of the magnitude of v1 as is the velocity direction 62. Variations of RZ/Rl were made with respect to a nominal R1 corresponding to an orbit altitude of 300 statute miles. This altitude was selected on the basis that it is above the high atmospheric drag region 2:200 8 mi) and below the earth's radiation belt 2:400 s mi). No variations inR1 were made since reasonable changes, from a rendezvous consideration, would probably be less than 200 5 mi. This variation would change the total radius by only a small percentage. Orbital motion is, of course, affected by the total radius rather than altitude. The selection of data presented in Table I shows the insignificant effect upon AWi/Vlv Vg/Vl, and.fih of varying Rz/Rl and v1 over the noted range. With respect to the nominal initial altitude of 300 3 mi, the ratios 1.005, 1.010, and 1.015 correspond to increases in altitude of 21.3, 42.6, and 64 3 mi. Although these variations are relatively small, the results obtained do indicate the lack of sensitivity to Changes in RZZRI. 29 TABLE I—RESULTS OBTAINED WITH VARIOUS RADIUS RATIOS AND RESIDUAL VELOCITIES ACp = 90°; 81 = 0°; ti/T = 1/4 (vz/vl) [62 Avi/v.L H v p% 1.005 1.010 1.015 1.005 1.010 1.015 100 .284 255° .287 254° .285 254° 1.297 1.500 1.500 (200 .285 255° .284 254° .285 254° 1.299 1.298 1.500 400 .282 255° .285 254° .285 254° 1.500 1.500 1.501 .A¢ - 180°; 61 = 80°; ti/T = 1/2 (Va/V1) (432 avi/vl R2 v1 1 1.005 1.010 1.015 1.005 1.010 1.015 100 2.128 47° 2.124 47° 2.124 47° 5.480 5.474 5.474 200 2.125 47° 2.122 47° 2.120|47° 5.474 5.475 5.470 00 2.115 48° 2.115 47° 2.115|47° 5.459 5.457 5.455 mo- 270°; 41 - 140°; ti/T - 5/4 (vz/vl) [42 Avi/vl R ; 1.005 1.010 1.015 1.005 1.010 1.015 v1 100 7.804|21¢’7.842|215’7.882 217° 11.004 11.049 11.099 200 7.836]213°7.833|21T37.813'2160 11.045 11.044 11.127 O l400 7-893Lélé.7oBZBl§gg§'7.978|215°,11.110 11.055 11.219 30 Further, the variation would, in many cases, include the altitude changes made during actual rendezvous. Since the trajectories under consideration are coasting arcs in a conservative field, a trajectory fromP1 to P2 is the reverse equivalent of the tra- jectory from.Pz to P1. This is illustrated by Fig. 4.4 in which the reverse quantities are denoted by primes. The following equalities are noted: ' ' v1 [£1 = V2 ‘53 0 .4171 " Avi VéLfié'vlL‘fll' Thus, except for the secondary effect resulting from a slight change in the reference (initial) radius, the data obtained for radius ratios greater than one also furnish data for the reciprocal ratios less than one. In order to study the effect of variations in transit time, a nominal time, Tn, was determined and. variations were taken with respect to this time. The nominal time was determined as the time corresponding to the case in which the perigee of the transfer ellipse coincided with the initial position. For this selected I a u... p .. . . ./ \ a 4‘ r .x \ ,r .. x /. ~ 1" ~‘ .1 /. .4 / ‘ . , x n / /. A . / \ .3. . /z./ 4 // 4 .. I / I . 1. / J4; .06. 4nd /1../ q n / (v/ v I! ‘4 ti; ‘ I. / 4’4, v 4.! 4” 4 III/t \ K . r l/ Ox ‘ . I. . . r- [1.44 It! a J . affix _ \ . \5 ’11. X ./. p \ 14...... x .1 _ l’l ., 14/. 5 . 4.....- w ...... )1 / ... l _ Ir (1 /. III II. cl) II: it ... .l ( v D RI? ‘I 0“ 4 I t: ’ /I/ w- e u |. 11 n/. 1 V i l/ 4.”. ‘ I .v e. ’1 , / 115‘. . r- .n 1. K 2 .I a...” . , . 32 R1 - R2 008 ACP After evaluating the other relevant ellipse elements from the orbital equations of Appendix A, Tn is deter- mined according to Eq. (A.1l) by setting TD = 0, which gives Tn = (E - e Sin E)/n. (4.21) Variations in transfer time up to T = Tn t 0.2 Tn were investigated. This magnitude of change in T is comparatively large. Transfer times of 0.8 Tn, Tn, and 1.2 Tn for each of the transfer angles A¢ *3 90°, 180°, and 2700 are shown in Table II. In the case of ACP- 90°, it may be noticed that the variations in the orientation of the transfer ellipses are such that the perigee is at P1 for T = Tn, the perigee is between P1 and P2 for T = 0.8 Tn, and the apogee is between P1 and P2 for T = 1.2 Tn‘ Table III shows the effect of transit time variation. Although results certainly do show the effect of time variations, fluctuations are relatively small for the time variations involved. The values of Mp, ti/T, and 31, selected for the preparation of Table III were selected to present a Wide spread of data. Primary Effect Parameters The variables having primary effect were found 33 TABLE II-—TRANSFER ELLIPSE ELEMENTS CORRESPONDING TO DIFFERENT TRANSFER TIMES A¢= 90° T-Tn T-OeBTn T'leZTn T, sec 1417.2 1133.7 1700.6 p, ft 22,755,110 28,551,458 19,502,997 a, ft 22,758,585 55,427,124 20,172,521 8 0.0100 0.3776 0.2076 m1, deg C 517.0 155.5 (02» deg 90 47.0 225.0 49’- 180° T . Tn T = 0.8 Tn T - 1.2 Tn T, sec 2849.2 2279.3 3419.0 p, ft 22,522,994 22,522,995 22.522,995 a, ft 22,525,554 25,499,807 25,029,011 8 0.00498 0.19316 0.13278 CH: dog 0 88.5 87.8 C2! deg 180 268.5 267.8 49’: 270° T'Tn T-0.8Tn. T-‘l.2Tn T, sec 4324.6 3459.7 5189.6 p, ft 22,736,110 20,491,523 24,409,270 a, ft 22,738,383 20,862,765 24,718,922 e 0.0100 0.1334 0.1119 4n, deg C 228.0 41.2 ¢Q, deg 270 138.0 311.2 34 TABLE III—RESULTS OBTAINED WITH VARIOUS TRANSFER TIMES Rz/Rl - 1.010; ti/T = 1/4; v1 - 100 ft/sec: ’31 - 0 (Va/V1) [fig 7 Avi/vl r ;::IE§ 0.8 1.0 1.2 0.8 1.0 1.2 ‘44’ 90° .292 257° .287 254° .285 250° 1.505 1.500 1.295 180° .210l25° .220 27° .244I52° 1.857 1.945 2.025 2709 .551 71° .795 71° .919 71° 2.507 2.481 2.550 Rg/Rl - 1.010; ti/T - 1/2; v1 - 100 rt/seo; finls 800 (ya/v1) lfi zznri/vl EL. £> n. 0.8 1.0 1.2 0.8 1.0 1.2 O 90’ 1.158)555’1.214 554 1.257l555° 2.554 2.845 2.217 180‘3 ;.804 47° 2.122 47° 2.285 45° 5.214 5.474 5.722 270’ .185 70° 2.588 59° 2.584)59° 2.275 2.414 2.505 Rg/Rl - 1.010; t1/T = 5/4; v1 - 100 ft/sec; .51 - 140° (Va/V1)L@_& Avi/Vl n 0.80 1.0 1.2 0.8 1.0 1.2 403 90° 2.505 44° 2.510 42° 2.451 59° 5.175 2.940 2.722 0 0 180° 5.152 159 5.075 159 2.950]171° 5.540 5.589 5.522 (270° 7.401121437.s41)217°8.251 219° 10.445 11.049 11.500 35 to be A45.61 and ti/T. A number of computer runs were made to determine the effect of these variables on the parameters Sv/v1,4y1/vl, vz/vl, and.62. The somewhat surprising discovery was made that the results showed a periodicity of 1800. with respect to .31. This is seen in Figs. 4.5-4.8 which show the following re- lationships: O B “/171 at (,61 + 180 ) 6v/v1 at )31 Avi/vl at (51 + 180°) = Avi/vl at .51 O vz/vl at (.61 + 180 ) = vz/vl at [91 £2 at (£1 + 180°) - (I32 at .91) + 180° . Figures 4.9-4.13 show the effect of varying M, and ti/T. It is seen from the curves relating 5v/v1 and I31 that a third impulse is potentially useful for re- ducing the transfer energy in the case of each transfer angle for which data is shown. However, on extrapolating plots of peak values of 8v versus Acp (Fig. 4.14) it is found that the curves intersect the abscissa at a value of 41¢ approximately equal to 30°, suggesting that no potential exists whenever the transfer angle is smaller than this value. Verification of this point would require more data than was obtained. In general, the potential increases with the transfer angle and becomes substantial at the larger angles. It is also noticed from Fig. 4.13 that the potential (1.))(Il).l~.ils.) .L)’.((.( )1) )I!)l,(..\ .) .14.). I 1.413)].(1): ( (1.11)] )r. 1 .4 ( . (( I. . , . ooopmoo Jatzoaeomfio 2.8324, 583% 0me _ owm, , .owm . cam com, 81.. . omg 1.1.5 om; _, .3, .51 o m 1 so :6 N . m w O W . {a m . mo 2 . 3 1 G . o; a O 1 L N; m m NE NE . I: m on n. M . ®.H I oom mmms u .H. 1 , 005m 4 sq s\m in u a)» o m H04. I .HENm t . Z . ,N.N - 8:858 sedans 8551.9: .82. . - 1 ., _ H C O 745 O 13) H , H H =9 1 / J9 m p :a 530 SOC 171‘ 5:5" 5'2”"IHIBD 175.2122 5 r 37 mmwhmwa .€.WTZOHEommHQ wBHUQHg A azaaqommmrrs.e .on H.. :n .. . .-1:( . _ _ . . 1 u . _ . _ u . 1 . _ .1 1 . . _ 1 . . . . . 1 _ . .1... _ 1..- ) v s . .1. J: 1. 1. .1 . .. . . A . lv 4......) .3); . 1.. 1b).!) we .L..(v.(.v)+1).x.|)l7.i .))..+).....I)u 1 . , .1 . 1 . . .. 1 . . _ _ , . . _ _ . ... n 1 . H . w _ 1. m 1 . I. ’. L . a . .. . .. I . .. 4 . L 4 .( . . Z 1 . . ~ 1 a _ g m _ . . . 1 . 1 . . 1 _ ((1)..--)1 1111(721i-.r-.-(.i--i.l--.111(1)).1 1 1 . 1 _ _ tr) .1 .. i) 1.).) 4 IIIII.. I I IIIWIIIIIIIIIII 1 _ . 1 1 II .1 11 1 .H 4 1 1 1 . .1 1 1 1. .1. . 1 1 1 II 1.III 1 1 I 1 . . _ . I 1 1 I _ .1 _ 1 1 - 1 I . - -.1- III 1 .1 1 1 1 1. 1. 1 1. 1 H 1 u 1 1. 1 1 1 1 1 . 1 . 1 1 11 1 11 11 U 1 . 1 1 1 D II . I III-IIII 1 I I 1 I III I I III“ a I I III III I1 II I I . I _ III IIIII .1 .I I III 1 I I I . . 1 . 1 1 . .. 1 1 11 1. . _ 1 1 1 1 1 .1 1. .. 11 1 1 1 # 1 1 H M 1 . _ _ _ _ 1 u _ . 11 1 III. 1 III I1 I I II. I .. III .. 1 I III I . | I II ..I. I. I I 1 I . III . I- 11 1 .11 . 1 1 1 1 1 1 1. 1 _ _ . . . 11 1 . u 1 1 . 1 . o . .. 1 .I y . . 1 I. . .+ . I 1 I . u . 4 v. ~ 1... 1I L 1 y . . ..- . “I. . r + 1 . 1 4 . c. I . .w. a . 9- .. ..IIM, 1 p . . my. 1 ~ . . h. 1* b. . «..Iw... 1 M. . 1 . I.III|I1|I I II III . I ..IIIIII‘ III I IIIIIIIIJIII -II In. .. -.I“- DI..-" I II"... I... - .0- I I t'\ (..I! 1' I0 .-I a,“ 3.60 o ‘ 6 I- \ ’ J I I . 1 . . . . 1 1 ._ a ,1 1 _ . .. _ _ . 1 _ . 11 ~ 1 1 o . . _ _ 1 . . - . 1 . 1 . _ 1 III.-. 1 . . . - .I- . -. 1 . - . 1 I . . ..I 1 _ -.1I 1 -I . . 1 1 1 . .. . _ 1. _ _ 1 . 1 1 1. 1 1. . 1 1. 1 1. 1 1 1 1 . .. . . 1 I II I I II . .II . ..II I p I I I IIIII . I II I . I I J.- II . I 11 .II “I I I . . 1 11 . 1 1 . w 1 1 1 . . . 1 . 1 . ¢ 1 1 u 1 n . ~ 1 . . I» I I 1 . 1 1 1 1 . _ . 1 I t . I . 1 . 1 . IIII.1I 4 _. o ..... . I III. III .. .4 a I I. .1 | . I I . II 1 11 II“ II . Y 1 . n 1 . . . 1 . _ m . V 1 1 .. . . . 1 1 . . H .1 M 1 .1 1. 1 1 w T? A1 I- I. . ¢ II. II»- 01+! '9 . III IIII » I Q ..1... IzolI II a I II . I ..I 1. . I I . I I I 1 . II I I .I. 1 .. .. - 1... I 1 _ 1 1 _ . . 1 1 . 1- 1 . . 1 . 1 1 1 1 1 1 _ . 1 1 1 . 1 . . 1 1 . 1 1 11 1 . . 14.. .a 1 .- III] IvIIIII“ . II II * II .I ~ .. I II I I . I II " . I - 1 - 1 . . . . 1...... - 1 . 1 1 «I 1 1 . 1 . . . 0 III. 4 - 1 1 . I WI] In I II .H.I.I .II._I..l.II..II . I III I . I . . I l .I I m _ . 1 .1 m . 1 1 1 1 1 . _ 1 1 . . . - 1 1 . 1 . 1. 1 . fl 1‘}?! \.9. 9......IIIW .I I . .I II 1 II. . II II o I 1*. '0 I I I II 1 .1 w. 1 . 1 I. o .. n a; _ _ . 1 1 1 1+ I. \ 1 1 1 . O. I If _ 1 1 1 . \ .f \ W 'I 1 I . 9 1 ulr _ .L I IIII. I III II -.III II II III . I - . . I I I II .. I V I1. 1 1I _ _ . 1. k 1 1 .- 1 1x . J 1 1 1 w $ _ vu. $1 T a . . . .11. 1.. ~ . . -1 1 1 . 1. 1 _ 1x 1 1. l I | . I 1 . III I .I.. I I. . I .- . I I l I or. _ m 1 . . 1 ‘H JIF L :1. . _ .1 1 _ 1 a . - . .1 \ . 1 . 1 J 1 . .. r .I w 1 1 1 _ 5 . I w . R r I ..I-I r I. r . .I I- I . I . I - I -. . 1. r. . II II I II I O I . ,4 .I‘ . s r 1 J 1 1 111 H u. H 1 1 .1 . J . _ I ~ r I II II. 6 I I I I II I I I I - 4 I I .I. . . I I . ‘ 1 a! _ 1 HI I, 1 t v 1 _ 11 4 _ w . I W 1 H W . 1 1 1 r _ . I . .r _ 1. 1 1 u II-- I.. LII-y I- - - .-I. - -I . - - .1 - -II . III I I- - -- EM -- -- ..I I . J 1 . fi _ . .1 . 1 _ 1 u 1 1 I ‘0 I‘. . 1H 1 1 1 1 1 1. e. - I . -. . I Q I.|II r IIII II I“ II I vII I fl I .I l, II I . II I v . . I ~ . 10 1% I _ 1 V I I. fl A. 1 I I.% / Nu 1i .I- .-I-II. I I--- I- I. - - I -III - -1 -. 1 .. I -II 1. ..... I 1 -- H . . J. . .1 H x 1 .1. 1 1 1, . , 1.. - - 1, 1 . 1 1 . 1 I . 1 1I II 1. - 1 II - 1 1 1 I - - .. 1 1 i .- 1 .1 1 u 1‘ . 1 m _ 1 1 1, . 1 ... _ _ M a . 1 1 . . 1 . 1 1 h a .1 § . . o}!I9.a 1 1 r 1 H . . O I . o . . I . 9-13. III/III . - I 1 -I .I III II \ .II. I - --I..- I117 . -: - I .II I I . II I 1.3 III - -II 1 m . v 1.- 1 . I ..I. f . . ..... I . 1 . .. . 1 u I u 1‘ _ 1“. . I. s .It. 1 1 . .11 w 1‘ \b '60 If ‘ 3.3.. . .10 r o \ II I. r ' I III urm fi I II x I | I . I II I I OJ 1 . . ‘ III 1 .t II I 1 _ r I V . . . I 1 1 1 . 1 . 1 J. 1 pl r )1, I» . 1. I 1 1 .J f 1 1 H Y1II 4 17.. r I. ‘L ‘ 1 0’ Ir _ I. .1! .. .I.. I..- --I-1 ELI I I -.I-I- I- I- I-I-. .1. :1. wk _-.I- -I- . u 1 . 1 1 11 f, 1 1 1 1. 1 m 1 .1.1 1 _ F M L T 1 w m . H J J16 Ifi .L _ . 1 . 1 Ito 1 1.4 fl ~11 1 . . 1 . fr 1 ‘ «o I . 1 I . . 1 :- ... 1 . . ..I-.-IIIQIQIJII'I‘II . _ I . . u I .0. I. :11? - .. rI . I II I . I _f. U . I. r 7 . ..Idik TV. 0 #1 . . . . .I. L 00. . . oII I.III . 11I.II .. a! '9. M r. .1.“ ..‘X. .1. o c . o a 0‘ o 6 [VII-a . .I.I.I Yin _ .J _ 1 ‘ .QI‘I‘IIN 1 . 1 . . . 1 r'lOII' PI- 1 ‘ ltvlo VIA II III.- 1 0 1 .14.... ~ 4 . r . . .DIo-vl. 7 I!- ~ L ..fI .I I II. . I . ..u . If. .1 1 . o .. 4.. ¢ 1 1 out I 1 1 :III 1 \~ 1 p furl“ . b E a . 1 I18 I F P r.. .0 -|§ . I r .1 I I 1 I V 1 1 1 4 4 4 4 I I . ~ . 1 . I . . I o o I . I I l a . . I. .1 . - 1 1 1 - -1 . 1 4 1 1 1 1 H .II 1 a I _ 1 1 _ . . . I . . . 1 II . I . 1 I. _ .I III III , _ -I, I III I . . I . I I _ 1 1 . 1 1 1 . 1 1 v .I . ..I I I I . III I I I III. I I I .I! . I I III I II 1 1 1 1 _1I ...1\ . .. III-III 11II. I I. 1 I I I I II II .0. I III II J1. V. o -| I II I II I I I I II I1I V II I II I) II I II I I V.- 1.. -.1. o p~ I I . L __1._.___-_ 39 mmmgmoo .Hm7(oneommHo weHooqm> Q¢bonmm owm omm 0mm ozm com oma omH om o: owm » omm :\H NQ 0: ow 0mm mmm: u e oowm u Sq Ho.H m Hm\mm mmmmz 92¢eqpmmm mo onaomquuum.a .on o: ow ONH 06H 03m 0mm saeafiea ‘an-NOILOHHIG ALIOOTEA LNVLTHSHH .. II 17:1. . .I I 41.! I I II III II I I .l I I .I I IIII II -._I. I II III III I I IIII7I II I JII!II .II. I... I d 7 Q _ 7 7 . 7 7 7 _ fl 7 7 7 7 7 7 I . I . 4 I . .77 sf 7. 7 7 7 2...47:}. ..I 7.7.7 .77.: . .7. I..:.I...7.;. 77. 777...... .7 .7 77.4.. 37. .7 ,7 .. 5... 77.7.7 7 _ 7 2 .I 7. . I I. I . II . I I III. I I .. I .I . . I . III. ~3§17.7;.. 4... {..I-:7 .. , .. 73:33-1. .717... .4 . 7 ..I .71» 7 7. .. I 7 7*. . 7 7 7 7 7 7 7 . - 7 7 .- 7 7 7 7 7 7 7 _ 7 _ 7 7 7 W _ I 7 7 - . 7 7 7 I I. -I .- 117 I». . 7 7 7 77 7 7 7 ,7 7 _ 7 77 7 . . . +J 7 M 7 . 7 7 I I4 I J 4 7 . n. 7 7 ..I ’fi. QJ I I I I I I I I I v II {7; - ..7..7f4n..o.- 7 5’ .77 yIl-tIOIo Q - A- . v}.‘ '07. 7 7 . . A .....I{.‘.. J 1"0440 11’! I . U I.'1.r iv. 7 v .uiIf‘avlL. . '{I 7 7 a! If.- ..I'ff’onlltt.a. r r. I . .IIDI'u'.u- I I I II I . II 7. . la .0 I .J r. 7. J i _ H . 7 'b-Drtlw JJ 7 r1.o.v1'...... ”.7 0d 7 j....‘ ¢G ..- J g 7. 7II Iv I III. ..7 4 II .7I r17... .7 I J. r 77 _ ’— 7 7 . r. 7 7 o I.) 7 | 7 7 . 7 7 I r . .7 . .. . 4. .7 . o . .. _ I. 7 .7 . .L Ir7 I r 4 I I I WI- a r. II I I I .7 H _ h H Ir «J 7 7 '7 .l I . .7 9 . I. 1 v4 . I . .u. r .7 c _7 I M 7 7 74 : . . .. 7 7 I I l I: 1.. O I III I I I x a I O .7 . m .7. D I; .7 II . . . I. ‘4 II ‘1. I .7 II .I M . - . . 7 7 H . 7 I I - u - _ .I u I III I 7. u . .7 7 ... 7 w .7 w I. _ 7 I I 7 a . 7 . - a 7 14141.... .T; I _ H I - 4 .v I I I I v 3:7 II IIII-I o 7' . .. I; 7 .. .I A . Ir .17. 7 ... u H I J [If 7 . I . 7 7 .7 . 7. 7 7 7 7 I 7 7. a 7 II.”- t «7 III I . III IL II. 1!}. I I. I I _ . 7 . ..7.II...-w w p 7 H .53.... I 7 I; I “7 I _ m u 7 ”D t .7 m . r . I . V 7 I IIIII I IIII I II A ..no II 7 I r I. o. IIIII I - I If , 77 . A 1 7 7.. . . 7. c; 7. . . 7 .1 . .7 J .I w I _. 7 ,- . . I7.- 7 7 . n 1 m J? 7 7 - 1.. 7. I .v . . 7 - .I I . 7 7 . 7. 7' p I . 7-JLI..I 7 I II. 4 . 7r II: 7..- . . I 7 737.7... _ 7 l .PVI$.'. .. 7 7 _ f4. - .7...r&lll'1 . . .rTIJI- I 7 . . . 7 7 .7 _ 4 Pic... 7 . I 7 . . 7 7.I>If.bl'.— . . . m w 7 _ f r 7f .aaI' 9147...... _ I . v .. 7 7 7 4. -I II III I. ..II I IIIIIII ' IIII7 u o I II 7 I - I II. J: I 7 7 . 7 . - 7 7 w . 77 1.94 7 I . 7 7 .1. - I _ . I r . 7 4 7 . u 7 7 I114. or 7 . .7 7 7 ..v‘III. ...p.. f 7 . ,r 7. .7 . - W . 77 7 .4 ‘7... 7? I _ .7 II I- I . IIII.I. . II I I I II 7 III. HLII I . 7 7 Q . . a r 7 - 7 H 7 . L 7 I 7 I III I .I II I I II: II D I II I I . W ..W a 7 7 . . 7 » 7 {I .- . . . . .. d .o. m 7 7 7 r 7 7 I 7I II II'II I» D b I II I V L 17‘ I 4 ‘ 7 . . . 77. 7 7 . w ..I . 7 7 7 _ .I 7 _ . .7 I .7 .- ...7 7 .. _ 7 .7 . I I I 7 I . III I . IJ II I . I 7 7 7 77 7. 77 7 7 7 7 _ 7 7 7 7 I 7 II III I IIII I II _ I I IIIII 7 — ~ . ’Id/ V IIYV - YO - '1 r V I 7 7 7 7 II I o A IDlIr- .I o I I 7. II.II 7. I . I . Jk 7 - II III I7 III III III I . III .I I 7 7 7 “.7 I.L 7 .J I7HIIU .1417 II 7. ..7 7 7.7 . 7LI.. 40 FIG. 4.9-—THIRD IMPULSE PARAMETERS Aw- 45°; Rz/Rl - 1.01: T - 700 890‘ 0.30 0.20 :_v Y a 3 001° 0 5 0 4O 80 120 160 200 180 220 260 300 340 31, Degrees 1.8 1.0/k 0.8 2‘ 0.6/ '1? w 0.3 ~ " '0 £0 80 120 160 200 180 280 860 300 540 61, Degrees 3 1/2 Ali. 1/4 x V1 m 1 . _, o I O 40 80 120 160 200 180 220 260 300 340 6.1, Entrees 80 40 360 320 62 Degrees 280 240 200 0 40 so 120 160 56?) 51, Degrees 41 FIG. 4.10-—TKIRD IMPULSE PARAMETERS qu- 900::R2/R1 = 1.01: T = 1417 See- o 4 o 120 o 40 so 1‘ 180 220v 250 300 340 180 220 260 300 s40 51 . Degrees ’31 , Degrees 19 120 15 80 40 $2 560 Degrees 320 280 p, .. 6 a 3” 7‘§* "w 240 “ ' | ’M, ,9 ; o 160 200 0 4o 80 120 160 29b 190 20 250 300 340 £1, Degrees 51, Degrees I} EHETEQAfiAq*Hé&B§HI‘qflIW@.~eIt; .01! ' "TIE???“T " V’eeh~?IA£ , T ;:o,; - Ixxgs :éoe -€&xw ‘ 7:;"i *ff! ‘1‘ 3.0 3‘, Imp ’\8 ' ?\‘J - * 1 1;v ‘ ' »sxo:»{?' " 3.0 #XI od~ «a; as: 65' o; ‘ o;r con oas 03$ 08; tmxfl .15. ,7 4 ‘ 7,9 7‘ ”H11 .‘ 5E1 05 0‘ ; “'WH“ . regesm 033 oat asses 42 FIG. 4.11—THIRD IMPULSE PARAMETERS ‘zitp - 180°; rag/a1 - 1.01; '1' -= 2849 Sec a 12 10 2 8 $1 1/T = 3/4 AV: 5 1 V1 1 4 2 ,‘ o o t 'J\l-/J 4.“ 4o :0 80 40 80 180 160 200 180 220 260 300 340 180 220 260 300 540 31, Degrees 131, Degrees if - 12 ' 2‘0 10* ‘1 ,' 200 160 2; & 120 '1‘: Degrees 80 ' _ 40 o . 0 1309 220 260: 300 340. 41,, Degrees 1" \ 1 O I f‘r' 43 FIG. 4.12—THIRD IMPULSE PARAMETERS A4? = 270°; Rz/er 1.01 T - 4325 See 5 z ti/T=3/4 51 4‘71 V1 1 o , i; 4 so 120 160 200 o 40 80 120 160 200 150 220 240 300 340 ,91, Degrees 51, Degrees 280 240 200 160 fig 120 Degrees 80 4O 8 6 0 0 4 180 880 260 300 340 51, Degrees 131, Degrees W . 44 FIG. 4.13-—THIRD IMPULSE PARAMETERS Aw: 315°; 122/111 = 1.01 T - 5257 See 0 180 220 260 300 340 /31, Degrees 3 2 1/T - 3/4 AV1 v1 1 1/4 '0 4 80 120 160 200 40 80 120 160 200 -180 220 260 300 340 180 220 260 300 340 : 131, Degrees .61. Degrees 280 240 200 160 52 120 Degrees 80 1/4 4 1/2 40 0 4O 80 120 160 200 40 80 120 160 200 51, Degrees .L- . . h a 12.1 d _. . . _ l..lv[...x\..!:64 , r ._ I I. "T““‘ f' .. _ ..L .l subwafiflf’ . _ -- L I .. “-1. DD :IQMPULsiE. _ . ., .. 11.. .61]. . h V + m . _ _ _ _ “11.. .1 m m * s _ . s m _ — — r; a. .UES : ITHIR 1.. .111le 01 :1 411111111 (19111.11 1‘10 "V‘Jflfm-u- 111 it .0111 7 6331313! V HQZ‘I 46 exists over a wide range of 51 when A0 =3150. This wider range would be expected as A¢>approaches 360°, since transfering through a large angle to a nearby point of different radius by applying only two impulses ‘would require a large expenditure of energy. This may be seen from Fig. 4.15 which depicts a vehicle in an initial orbit A. By the two-impulse scheme of transfer, one impulse would be applied at P1 to establish a coast- ing trajectory to P2, at which point a second impulse would be applied to match the target velocity. Traject- ory B of the designated figure illustrates transfer by this method, and as shown, a large change in velocity direction would be required at P1. A much less expen- sive method of transfer would be to let the vehicle continue in its orbit until it reached a point near P1 where an impulse could be applied to place it on a near Hohmann transfer ellipse (orbit C) to P2. It is not meant to imply that this latter method of transfer would minimize energy, only that it would obviously require less energy than a direct two-impulse transfer. Sample Problem As was stated previously, the existence of a third impulse potential does not ensure that a third impulse can be profitably utilized. It is necessary to also con- sider the initial and final velocities. The following problem illustrates the usefulness of the presented data curves for determining the applicability of a third im- pulse for an assumed set of conditions. 47 Pl-Pz: TWO IMPULSE TRANSFER Pl'Pi'PZ: THREE IMPULSE TRANSFER FIG. 4.15-—COMPARISON OF TRAJECTORIES FOR LARGE ANGLE TRANSFER 48 Let it be assumed that a vehicle is to be trans— ferred from.P1 to P2 where the conditions of transfer are as follows: R1 - 22.511 x 106 ft (300 e mi altitude) R2 - 22.73611 x 106 ft (342.6 e mi altitude) Ago - 270° T - 4324.63 Sec ' 72.06 min v - 211 ft/sec at 1200 0 v1. - 253 ft/sec at 102.20. The velocities V0 and vf are given relative to a coast- ing axis system as described earlier. A coasting trajectory Joining P1 and P2 with transit time as noted above would require absolute velocities at these points as follows: Vi (Tangential) - 25,132 ft/sec Vi (Radial) - 0 V2 (Tangential) - 24,884 ft/sec V2 (Radial) - ~249 ft/seco A two-impulse rendezvous could be achieved by simply applying impulses to cancel v0 and Vf and would require e characteristic velocity of AV= v + v - 464 ft/sec. o f In order to examine the applicability of a third impulse, plots of Vi/vl! vz/vl, and ‘52 versus 31 were used to determine the characteristic velocity required for rendezvous via three impulses. Only the case in which the magnitude of VI is equal to that of V0 was 49 examined. Thus, for v1 - o - 211 ft/sec and for various values of461, the quantities Ayi, v2, and 6% were found and the characteristic velocity determined from the equation AV - [v1 - vol + (Avil + Ivf - vzl . The solid curves of Fig. 4.16 are plots of.AN versus .61 for various times of application of the intermediate impulse. ‘Minimum values of these curves are Joined by the dashed curve. On extrapalating by means of this dashed curve, the minimum characteristic velocity required for a three-impulse rendezvous is found to be 200 ft/sec for an intermediate impulse applied at a time between 3/8 and 1/2 T. Thus, by using three in- stead of two impulses for rendezvous under the stated problem conditions, characteristic velocity is reduced from 464 to 200 ft/sec. .111.1A .1.. 1 1 1 1 1 . J ..H1 1 1 4-1 . 1 . 1 1 . 1 1. .1; -1 1 .., 4.x . 1 LL. 1 , . -—.‘b_‘ 1 -.1- l . w 1 1 1 l I .1;1~ . 1 ’ 1 l . . . , I ' 1 L ..._-,_... _ i 1 1 l 1 1 1 11.16. 1.21 ‘ ‘1” 1 1 1... 1 .._._.. 1 1 1 1 , 1 1 5O ._11_. 44-1-"- VELOCJTE W1 ” 1,}Dégrse 111. ; ; 1_ _. 'c. --_,_- 1 1'1 ..11 A1 . 1 - ,1- :4.m .3 . . .-. - _ -. I . mm.. m _ H #1 u w . m . 11111 .11. 11 1 l 11:11 .... 1 - 1. 1 - 1 1.1.1 H. mm" .1“ _ m _ 1 U . m . . 1 .. . 1. .l. . 1 1 3... .N . 1.. 1 _ .m... 1 i . u _ i 1 mw 1mm _ 111,111.11 (1| : E I 1 1 1.1 A .411 1T .1 n 1U1 1 1 A. 1.11 p.11- .-._nu..1.vW.... . . ..1 _ -.w- .. 4.1 I {01.1 . ._ HM”. _ t . . . _ l h . H. 11.31 W-.. .A....l ..1.1”..1.w.1-e . .11—1111 1.1.1-1 .. it .1 - 1 .. . 1 1... 1- 1114 . mgr _ I .m .h u . v . .mv . I A _. «AMI. l ...UR. 1713...: 1* 1 L- .11 .1. _. . 1.01 .1 .2m.. 1 _ 1. H , m _ m .., u . . . 1 . fl 1 - . I : -111 11.. . 1“ fl.-. .. 0U... 1...- 1 . :3... D11 “ ,_C. _ . , M." .1 . T. . l 1...“..4. . .511.» . El 1 .. .. . . .. ._ r. m_ ,mm. L _ 1. I 6. ._ g. TENT - 11 . @11111 _ .. 3“ 1...“. M. w I .1. m ..I... M M ,H l 1. 17_g.. . 1. 1 up. u . ..r .W. 1111.11 1.. ..... 1r 1. 11.1111 '1" 1} 95‘" 1 1 " 1.1:; 1 1 1 ..rw .. . ~_,_._ . Hun—— .. —~_~ 1‘;ng ;, .. 1 .... ' " .,. ~‘Ir. . 731;; .. .: L3 »)—_:1; ’I‘FE’W‘JIV “IT II! 3'1”qu $411391}; aafiEI'Ih—BL 41 9I3 M33 023 343mm 53933251115011 ‘7 T\ri. 1 “\5 . 7 g ./ . .7 .I c; 1: '1 “’. . '1“; 1717111 . :7. ’1 1 g i 11-rj 1:.117 g ’“¥:-11 .” 1.” 7:77..” 5" 1 1 ‘ 71' 1 7:11: 7 1~1 7». : 1 1 3' a ii ” 1 1”: 77f 1*31 u ‘1- . +171. . w ~g;m117 ~ i - m :1 rrtifll1: 1 111’g7 1‘ 1:11 1 74m. +1 3:3: -7 . 1159; 09;. a9 1 fli_.'1;fl I97 31.711:1117111:11:;1+ ;11 ,Ec»1.,:41 : ‘. . .::~ 13; 3545111393. Ignomrnmu mwmv1;..mwaraaa' . CHAPTER v SUMMARY AND CONCLUSIONS The problem of minimizing the energy required to perform.a space rendezvous has been analyzed by two different methods: the calculus of variations, and a trajectory perturbation technique. A brief description of each method and a summary of the results obtained by their application are presented in the following ‘paragraphs. CALCULUSIQ§E§RIATIONS The calculus of variations analysis (Chapter III) is based upon the use of linearized relative equations of motion which are shown in Appendix A to be reasonably accurate in accounting for the acceleration forces. From the set of Euler-Lagrange equations, two Lagrange multipliers are determined which are identically equal to the direction cosines of the thrust vector. It is shown that these multipliers are independent of the remaining set and can be expressed in terms of second order differential equations identical in form to the equations of motion. As a consequence of this results, and with reservations according to the assumptions made, it was possible to reach the following conclusions: (1) A minimum energy trajectory contains no arc flown with an intermediate level of thrust. (2) If the upper bound of thrust is large so that the assumption of velocity impulses is valid, 51 52 a minimum energy trajectory is achieved with either two or three impulses. .No criterion was established that would determine which of these modes should be used. TRAJ'ECTORY PERTURBAT ION The perturbation technique is based upon determining the effect of perturbing a vehicle relative to a nominal coasting trajectory. If the vehicle were to move along the nominal trajectory it would traverse a path between space terminals P1 and P2 in a time T. However, by the perturbation technique, it is given a velocity impulse at P1 which causes it to deviate from the nominal trajec- tory. Another impulse is applied at an intermediate time that brings the vehicle back onto the trajectory at P2 when t - T. A third impulse potential is defined as the sum of the relative velocities at P1 and P2 (measured with respect to an axis system.whose origin is restrained to move along the nominal trajectory) minus the inter- mediate impulse. The following results obtained from a parametric study pertaining to the third impulse potential, 5v, are noted: (1) Small variation of the radii ratio Rz/‘R1 and transfer time T were found to have only slight effect on 5v. (2) The intermediate impulse ANi and the terminal ve- locity v2 were found to be directly proportional to the initial perturbing velocity, v1. Also, it was found that.4%, the direction of v2, is 53 independent of VI. (3) The primary effect parameters were found to be .61 (the direction of the disturbing velocity), t1 (time of the intermediate impulse), and A? (the transfer angle). Variation effects are shown in Figs. 4.5-4.14. (4) A third impulse potential was found to exist for a wide range of conditions, with the greatest potential‘occurring at large values of (sq). How- ever, the results suggest that no potential exists for values ofupless than 30°. For a target in a circular, 300 5 mi orbit this would corre- spond to a rendezvous time of approximately eight minutes. The investigation reported in this thesis did not determine a complete answer to the rendezvous energy problem by any means. However, it is felt that much insight has been gained and that tools for further investi- gation have been developed. CHAPTER VI SUGGESTIONS FOR FUTURE STUDIES EXTENSION 9;; DATA The data presented in Chapter IV relative to the conditions under which a third impulse potential exists need to be expanded., In particularly, it would be de- sirabha to obtain data for larger values of RZ/Rl° In- sofar as transfer time is concerned, it is suggested that the times corresponding to the four transfer ellipse orientations obtained by placing the perigee and apogee alternately at P1 and P2 might be of interest. PRACTICAL COMPUTATIONS The method used to solve the sample problem of Chapter IV, although illustrating the principal of three impulse application, would not be practical for use in actual satellite interception. A method subject to rapid solution by digital computer would be needed. One method that would meet this requirement can be derived by deter- mining equations to describe the curves relating ANi/vl: v2/vl, andneé to.di, and making use of the ordinary method of maxima-minima determination. The parameters Ami/v1, vz/vl, and.62 are described with reasonable accuracy by equations of the form 2 (Va/V1), (Avi/Vl) = A "' B Sin /3 +91) (6.1) 54 55 Tan (52 + 5‘2) - £7851: £ng (a?) ‘1) (5.2) where A, B, C, and D are constants for a given value of AV and ti/T, and the 4 are phase angles. The accuracy of the describing equations is shown in Fig. 6.1 for the case in which 490- 270°, and ti/T - 1/2. The solid curves are actual values while the dashed curves were determined by Eqs. (6.1 and 6.2). For a fixed value of Ag» the constants and phase angles in these equations would depend upon ti/T; and their mode of dependency should be determinable from plots of the quantities versus ti/T. It will be assumed that the relationships could be determined in an appropriate form. The characteristic velocity required for a three- impulse rendezvous is determined by Eq. (4.1), which in algebraic form is AV a [v02 + v12 - ZVOVI COS (250 '31)]1/2 (6.3) 2 2 + v:l + [v2 + vf - 2v2vf Cos (.62 ~Er)]l/? On normalizing with respect to v1 this equation takes the-form (Av/v1) 3 [(vo/VI)2 + 1 " Zvo COS (£0 -61)]1/2 +A'V1/Vl '(604) + [(vz/vl)2 + (vf/v1)2 1/2 - (2v2vf/v1) Cos (62 -.ef)] 57 By substituting Eqs. (6.3 and 6.4) into this equation after having determined the quantities A, B, C, D and S‘as functions of ti/T, the characteristic velocity, .AN/vl, would be dtermined as a function of the initial and final conditions and time ratio ti/T. Thus, the conditions under which AN/vl is minimized could be determined by solving the set of equations egévgm = O (6.5) 29431 9(AV/v)! = O. (6.6) 29 ti T It should be possible to obtain a rapid solution to these two equations with the aid of a digital computer. It is believed that rendezvous studies in the direction indicated above would result in the development of a guidance tech- nique that could be applied in actual practice to reduce energy requirements. An alternate approach would be to develop a vehicle- borne computer to solve the set of twenty-seven equations determined by the calculus of variations analysis of Chapter III. BIBLIOGRAPHY l. Houbolt, John 0.: Problems and Potentials of Space Rendezvous. Preprint of paper presented at the International Symposium on Space Flight and Re- Entry Trajectories, organized by the International Academy of Astronautics of the International Astro- nautical Federation, Louveciennes, France, June 19- 21, 1961. 2. KingAHele, D. G., and Merson, R. H.: Satellite Orbits in Theory and Practice. Journ. Brit. Interplan. Soc., Vol. 16, 1958. 3. Lawden, D. F.: Inter-Orbital Transfer of a Rocket. Journ. Brit. Interplan. Soc., 1952, Annual Report, pp. 321-333. 4. Lawden, D. F.: Minimal Rocket Trajectories. Journ. Amer. Rec. 300., Vol. 23, No. 6, 1955, pp. 560-367. 5. Lawden, D. F.: Optimal Trajectories. Radiation, Inc., Special Report no. 5, RR-—59-—1186-—7, May, 1959, or Air Force 33(616)-—5992, Task 50861, R1 Project 1186. 6. Lawden, D. F.: Optimal Powered Arcs in an Inverse Square Law Field. ARS Journal, Vol. 51, no. 4, April, 1961, p. 566. 7. Leitmann, G.: Extremal Rocket Trajectories in Position and Time Dependent Force Fields. American Astro- nautical Society Preprint (61-30), presented at 7th. annual meeting, Dallas, January 16-18, 1961. 8. Leitmann, G.: On a Class of Variational Problems 58 59 in.Rocket Flight. J. Aero/Space Sci., vol. 26, pp. 586-591, 1959. 9. Kelly, Henry J.: Gradient Theory of Optimal Flight Paths. Presented at the ARS Semi-Annual Meeting, Los Angeles, May 9-12, 1960. 10. Bryson, A. E., Denham, W. F., Carroll, F. J., and Mikami, K.: Determination of the Lift or Drag Program that Minimizes Re-Entry Heating with Acceleration or Range Constraints Using a Steep- est Descent Computation Procedure. ‘IAS Paper no. 61-6, presented at the 29th. Annual Meeting, New York, Jan. 25-25, 1961. ll. Seltzer, C., and Fetheroff, C. W.: A Direct Varia- tional Method for the Calculation of Optimum Thrust Programs for Power-Limited Interplanetary Flight. Astronautics Acta, Vol. VII/Fasc. 1, 1961. 12. Miele, Angelo: A Survey of the Problem of Optimizing Flight Paths of Aircraft and Missiles. Paper no. 1219-60, presented at the ARS Semi-Annual Meeting and Astronautical Exposition, Los Angeles, May 9-12, 1960. 13. Hohmann, W.: Die Erreichbarkeit der Himelskorper. R. Oldenbourg, Munich, 1925. 14. Hoekner, R. F., and Silber, R.: The Bi-elliptical Transfer Between Circular Coplanar Orbits. Army Ballistic Missile Agency, Redstone Arsenal, Report no. DA-TM-2-59, January, 1959. 15. 16. 1'7. 18. 19. 20. 21. 60 Ting, L.: Optimum Orbital Transfer by Impulses. PIBAL Rpt. 636, January, 1960. Ting, L.: Optimum Orbital Transfer for Several Im- pulses. Astronautica Acta, vol. VI/Fasc. 5, 1960. Breakwell, J. V.: The Optimization of Trajectories. North American Aviation Report no. AL-2706, August, 1957. Moulton, F. R.: An Introduction to Celestial Mechan- ics. The McMillan Company, New York, 1914. Clohessy, W. H., and Wiltshire, R. 8.: Terminal Guidance System for Satellite Rendezvous. IAS Paper no. 59-95 presented at the IAS National Summer Meeting, Los Angeles, June 16-19, 1959. Eggleston, John M., and Beck, Harold D.: A Study of the Positions and Velocities of a Space Station and a Ferry Vehicle and Return. NASA Technical Report R-87, 1961. Goodman, T. R., and Lance, G. N.: The Numerical Integration of Two-Point Boundary Value Problems. Mathematical Tables and Other Aids to Computation, Vol. X, No. 54, April, 1956. APPENDIX A EQUATIONS OF MOTION AND ORBITAL MECHANICS The basic equations used in the rendezvous energy study are presented in this appendix. No derivations are presented since they may be found either in well known mechanics texts or in current literature. Deri- vations are presented in the noted references. ORBITAL MOTION The polar equations of motion of the center of mass of a satellite orbiting about a spherical earth are R - R¢F . aq/R21+ AR (A.1) Rd) + and: AT (A-Z) where (-LVR?)is the instantaneous gravity force per unit mass, and AR and AT are the radial and tangential accele- rations due to thrusting forces. In the absence of thrusting forces, the center of mass will describe an ellipse, parabola, or hyperbola accordingly as the sum of the kinetic and potential energy is negative, zero, or positive. The subject investigation was restricted to an anal- ysis of elliptical motion. The various parameters shown in Fig. A.1 which are used to describe elliptical motion are related as follows:18 semimajor axis - Ra ' 1/2 (Rapogee I Rperigee) (A‘g) 61 62 ‘7:— IE 8 £111 PE) PERIGEEI RADIUS RADI w 90 ‘v FIG. A.1-ELLIPTICAL ORBIT GEOMETRY 63 eccentricity - 9 '-Rapogee ' Rperigee Rapogee * Rperigee semilatus rectum - p - a(l - e2) angular momentum - J = (“ml/2 average angular rate - “A; 33 2 orbit period - P - 2fl%n radius - R B p 1 + eTSoscp eccentric anomaly - -1 E = Cos [(a - R)/ea] time from perigee passage - t - T = (E - 6 Sin E)/n perigee flight path angle - r -- Tan'1[(e Sin w/(l + e 008 W] (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) (A.10) (11.11) (A.12) 64 total velocity - V'= [Adz/R - 1/a)]1/2 (A.15) radial velocity - R = v Sin r (A.14) 01‘ ° 1 2 R - WP) / e 31:14) (A.15) tangential velocity - angular velocity - ¢ =- J/R‘2 (A.1?) or (i: - (v Cos n/R . (A.18) EQUATIONS 92 RELATIVE MOTION In order to facilitate the study of the space ren- dezvous problem, it is at times advantageous to use relative equations of motion. A convenient axis system is one which has its origin affixed to the orbiting tar- get. Such an axis system is illustrated in Fig. A.2 which shows a right-handed rectilinear system with the negative y axis extending through the center of the earth and the x axis in the orbit plane. The equations of relative motion arelg’zO SE - (y + Rx'é - 2(3} + fax}; - x932 (A.19) {-qx/fls - Ax CIRCULAR ORBIT EARTH SURFACE .4). EARTH CENTER FIG. A.2—RECTANGULAR COORDINATE SYSTEM I,” ll. Iii|-i I, , M..__:....- ..+._ . --m, ‘n ‘- --~“ O ‘\ . .\ H. . .\ _ fi \ . n a . 1.. r .. g.\ c o. s t r . . ... t a o In"! .1 . (4.!" .l'bu‘q. I- ii'i 'Iil“ alllli‘i\!‘ . W .. I. t It) ‘1Tl‘ ’1 if iiiog Ia. 66 y + xgo + zip.» ii - (y + R)¢2 (A.20) +lqjy + R)Ao5 - Ay. 'z' +uz/fi3 .. A Z (A021) where 1° - [x2 + (y + R)2]1/? The quantities Ax! Ay, and A2 are thrust accelerations. For the case in.which the origin of the axis system moves in a circular orbit and relative distances are not too great, the above equations may be linearized. (Actu- ally either of two modes of usage are possible: the origin is affixed to a target moving in a circular orbit, or neither the target nor vehicle are in circular orbits but their motion is expressed with respect to an x-y axis system whose origin moves in a circular orbit.) Assuming a circular orbit leads to a constant value for R and «3 '00 where, following custom, to is used to denote a con- stant value of {5. Then, in order to linearize the equa- tions,/°'3 is expanded as a power series and all terms of second order and higher are dropped to give #5 z (WRSHI-Sy/R) Further, for a circular orbit AVE; 8“? so that 67 On substituting this expression into Eqs. (A.19-A.21) and dropping the terms containing y/R which do not cancel, there results " - 2&4? - Ax . (A.22) §+2<.:i-owz-Ay (A.23) ,, 2 2 +002 - AZ 0 (A024) When the thrusting accelerations are zero these equa- tions can be readily solved to obtain x - 2[(2ibflu» - 3Y0] Sin‘dT - (2&0Au» Cosch + [6yo - ado/concur + x0 + 2&0/0.) (11.25) y 3 [(210/00) - 3Y0] Cosodr + (yo/in) Sinodl' + 4yo - zinc/w (A.26) z =- 20 Cosz + (EC/w) Sincdl‘ . (A.2?) Equations (A.25-A.27) can be used to determine the velocity components that would be required at a given initial position in order to place a vehicle on a coast- ing path that would intercept a designated target after a specified time. Assuming the target to be at the origin of the axis system, the required components are i0 ' xo SincuT + yo[6wT Sinqu - 14(1 - Cosch) A/w (A.28) 5'0 ' 210(1 - Cosodl‘) + 331(4 Sinw'I'm- our Cosz) AT“ (A.29) zo ' :39— (11.30) 58 where A= SUI' Sinwl‘ - 8(1 - 003m)- The linearized equations are reasonablv accurate provided the relative distance is not too great. For example, assuming distances of x = y = 50 statue miles, the discarded terms would amount to an accelerating force of approximately 0.01 ft/secz in the x and y directions and 0.0002 ft/secz in the z direction. It should be noted that the weak coupling between the z motion and motion in the x-y plane as seen in the exact equations no longer exists in the linearized equa- tions. For this reason, many investigators have chosen to analyze only the more complicated x-y (coplanar) mo- tion with the suggestion that the total motion be deter- mined by analyzing the z motion separately and superposing the results on the x-y motion. APPENDIX B NOMINAL TRAJECTORY EQUATIONS The method of Gauss can be used to determine the elliptical elements of an orbiting body when consecu- tive values of the radius and are swept are known with respect to time. Following is a derivation of the 1 necessary equations. Equation (A.1?) can be rearranged to give the areal rate being swept by a radius vector as 1/2 R(R 51%) = 2 dA . J= (Mp) (B.1) where dA/dt denotes areal rate. On integrating Eq. (B.1) over the time of observation the following equa- tion is obtained Asector - meL 'LT (13.2) The area of the triangle between radii R1 and R2 is given by the equation Atriangle = P332 5111155" (3-3) where Ap is the angle between the radii. The method of Gauss depends upon the ratio of these areas, which is I) - [Owl/231' . (13.4) RlRZ Sin Ago Upon substituting values of R1, R2,¢p1, and ¢h into 69 70 Eq. (A.9), two equations result which can be solved to give . ~ (B.5) p(R1 +R2) = 2 + 2e Cos ((92 + (91) Cos (¢2 -cp1), m slag 2 2 Through the use of Eqs. (A.9 and A.10), and after sev- eral equation manipulations, the relationships (8.6) e COS ((92 +CPi) _p COS (E2 - E1) (R1R2)Il2 2 " COS (sz " g) 2 e Cos (E +>E ) - 008 (E - E ) 2 2 1 'Z‘Tl (13.7) - (R1R2)1/ZCOS “Pg “@1) a 2 can be obtained. Equations (B.5) and (B.6) combine to give 2 (B.8) p ._. 2R1R3 Sin (4072) R1 + R2 - Malawi/zoos (Am/2) Cos (AB/2) where 4