On the Solution of Lambert's Orbital Boundary-Value Problem.pdf-资料库

~AFILE UNLIMIT'ED TR 38027 t 1h 107255 00 O ROYAL AEROSPACE ESTABLISHMENT Technical Report 88027 April 1988 ON THE SOLUTION OF LAMBERT'S ORBITAL BOUNDARY-VALUE PROBLEM by DTIC t ELECTE R. H. Gooding NVI98 otilfJ0TS QIJUYTEL) ARE~ K0 1 NECESSAFA.~ AVAW.ABLfl TO MEMBEms OF THE PUBLIC r1A TO commrpr[AL OR(,~I~tJ Procurement Executive, Ministry of Defence Farnborough, Hentr t . Approved f-pubdkmhc.e Dtndmutbd Um&inted UNLIMITED'S 8 11 01 0,28

UNLIMITED ROYAL AEROS PACE ESTAB LI SHMENT Technical Report 88027 Received for printing 6 April 1988 ON THE SOLUTION OF LAMBERT'S ORBITAL BOUNDARY-VALUE PROBLEM by R. H. Gooding SUMMARY During the past 30 years there has been a resurgence of interest in classical orbital boundary-value problem of Lambert, largely because of its relevance to space rendezvous and interception. The most notable contribution to in 1966, but more recent the subject was by Lancaster, Blanchard and Devaney, researchers have failed to build on that work; the present Report is aimed at remedying this neglect by providing details of a universal solution of Lambert's problem based on the approach of Lancaster et al. presents starting formulae for Halley's cubic iteration process, used for evalua- tion of the unknown parameter, always gives highly accurate values of x after three iterations. , at the heart of the approach; this process the In particular, the Report x A Fortran-77 computing procedure for a general solution of Lambert's pro- blem has been developed..adl,.its three main subroutines are listed. Details are given of the testing of'this procedure. Much of the Report is devoted to a classification of the set of all Lambert problems, and to a discussion of various geometric and physical aspects. Departmental Reference: Space 670 SCopyright © Controller IJMSO London 1988 UNLIMITED

2 1 2 LIST OF CONTENTS INTRODUCTION CHOICE OF PARAMETER FOR THE ITERATION VARIABLE 2.1 and L-s.milarity Lambert's theorem and the relations of L-congruance 2.2 Lambert-invariant parameters, Lambert-equivalent problems, and the parameter x 3 THE ALGORITHM FOR COMPUTING At 4 5 6 7 8 9 ITERATION PROCESS STARTING FORLMULAE Introductory remarks Single-revolution starters 5.1 5.2 5.3 Multirevolution starters COMPLETION OF CONVERGENCE COMPUTATION OF VELOCITY TESTING RATIONALE AND RESULTS CONCLUSIONS Appendix A Gauss's method and the paper by Battin and Vaughan Appendix B Details of the TLAMB algorithm Appendix C Fortran-77 subroutine TLAMB Appendix D Fortran-77 subroutine XLAMB Appendix E Fortran-77 subroutine VLAMB List of symbols References Illustrations Report documentation page QRAAI A-asion P07 NTIS DTIB TAB Uuaruouso ed Jualfti.tlon Q ~Distr ibution/ ,hv7$ ai i.1tt Codes MaY ~o D.t iSpuolal 3 5 5 7 9 14 15 15 16 19 21 22 24 28 30 34 40 42 44 45 47 Figures 1-6 inside back cover

:3 1 INTRODUCTION The 'orbital boundary-value problem', constrained by two points and an I elapsed time, is usually associated with the name of Lambert1, though Euler had studied the problem some 20 years before Lambert (but only for parabolic orbits); other celebrated mathematicians whose names are associated with the problem and ".. its solution include Gauss and Lagrange. Thus it is a problem of classical celestial mechanics, and one that (like the solution of Kepler's equation) con- tinues to attract the attention of mathematicians searching for solutiion pro- cedures of ever-greater generality, accuracy and efficiency. Good text-book introdu.tion& - Lu be teund ill l.Lj 2 to 4, whilst Refs 5 to 26 nre studies, chronologically listed, frtom the last 30 years; the outstanding paper on this list, though from as far back as 1966, is the one by Lancaster, Blanchard and Devaney 8 . Classically, Lambert's problem arose as a core component in the determination of an orbit from three observations of direction alone, the central observation being used (on a trial-and-error basis) as a source of the missing distance data for the other two observations. In the Space Age, with direction measurement a commonplace, the solving of Lambert's problem is directly appli- cable to the important subject of orbital rendezvous. Lambert's problem may be stated as follows: an (unperturbed) orbit, about a given inverse-square-law centre of force, C say, is to be found connecting two given points, P1 and P 2 t with a flight time At(G t2 - t ) that has been specified*. The problem must always have at least one solution and the actual number, which we denote by N CPIP 2 of generality, that At > 0 and the value of at - it , depends on the geometry of the triangle is assumed, for convenience and with no loss To get an immediate feel for the problem, let us suppose first that the * triangle CPIP 2 at C is not degenerate, so that 6 , the angle subtended by PIP 2 , lies between 0 and i . Then it would appear there must be at least two Ssolutions, since an orbital path (in the plane CPIP 2 ) can be found that subtends an angle 2rr - S (ig going the 'long way aroand') as well as one that subtends e . We can avoid this duality, however, by supposing the Uirection of motion to be specified in advance, so that the two angles can be deemed to define different problems. There is a further complication, since if At is large enough, other paths (necessarily ellipticct) will be possible, each of which includes a number of complete revolutions. It turns out (and will be apparent when Fig 2 is SN 0 * A list of symbols is provided at the end of the Report.

4 introduced) m say, normally occur in pairs; thus as At that paths incorporating a specific number of complete revolutions, increases, N (for a given triangle and specified direction of motion) even (instantaneously) at each (critical) value of At at which emerge, coincident for that precise value of At multiple revolutions by extending the distinction between individual Lambert problems so that '0' is regarded as an angle of unrestricted positive magnitude is an increasing odd integer, apart from being two new solutions . We simplify the approach to defined by the geometry of the path and not just by that of the triangle; we write ar for the reduced angle (such that to discriminate. would imply negative At which we have already excluded for convenience.) We (We do not have to consider negative 0 < Ur < 2m ) when , because to do so is necessary it 0 have effectively redefined N such that N = 1 if other hand, N critical value. 0 < 21 ; , depending in the relation of At 0 0, 1 or 2 G > 2P , on the if to the appropriate Turning to degenerate triangles, we consider these on the basis of so that B is now k'a for some the unlimited values of 0 just introduced, integer (ý0) k . Then if orbital paths exist that are not rectilinear, their number must be infinite, since any plane through the degenerate triangle contains valid paths. If we choose an orbital plane (as well as the direction of motion) arbitrarily, however, we have N = 0, 1 or 2 is odd, exactly as in the last paragraph; this is actually the simplest of all cases to deal with in prac- tice, though the literature contains a number of solution procedures that fail here quite unnecessarily. But there are real difficulties when (- 2m), associated with a type of discontinuity that is described in section 3. is even , if k k The effect of this discontinuity is the angle (kw) , which symbolizes the representation of 0 as 2(m - 1)r that we would like to be able to distinguish plus a 0r of 2v , from (kr)÷ , which symbolizes its representation as 2mn plus a 0 of zero; if (for an arbitrarily chosen orbital plane) N r this distinction (or an equivalent one) 1 1 or 3 if m is not made, 1 , and then N = 0, 1, 2 or 4 is even) unless if m > 1 . The orbital path has to be rectilinear (when k P and P2 coincide. We can now summarize the data involved in the solution procedure to be the present Report. The input qucantities are the constant o developed in (strength of the given force centre at C ), CPl2 ) there is also a Lambert problem when m < 0 hyperbolic solution, wholly internal to the triangle. The transitional case, (the unrestricted angle PICP2 ) and At r 2 (We assume m > 0 (equal to CP1 and , but r1 and . 0 < w , there is ; if 0 , then a unique

5 , is of course trivial, but even this can be treated as a Lambert 4N + I in number, consist of N itself and (radial velocity) and VT is assumed here that values of 6 equal (transverse The output quantities, with pi 0 problem.) N sets of four quantities, vie VR velocity) at both P1 and P2 . It to (2mw)÷ (2mir) and can be distinguished, so that N does not exceed 2. In reality, of course, the real-number system does not permit this distinction, though this is a somewhat academic point in a computing procedure that can only operate for the finite set of computable numbers; more importantly, with 'multiple revolutions' of a rectilinear orbit the problem has become completely academic anyway, since it involves at least one infinite-velocity 'bounce' off the force centre. Nevertheless, we shall find it advantageous tu compute with a peir of quantities, q (introduced in section 3) and ma , in place of just e ; this avoids the academic difficulties, and is also more efficient. As with Kepler's equation, Lambert's problem has no satisfactory direct solution - we have to proceed by an iterative technique (trial and error) and this inevitably dominates the solution procedure being developed. The following issues then arise, and will be discussed in successive sections of tho Report: first (in section 2) the choice of a suitable parameter of the motion to use as the iteration variable x the problem is is sometimes claimed, for example in Ref 9, that inherenzly a two-parameter problem, with simultaneous iteration (it needed on both parameters, but this claim is unwarranted); second (in section 3) the 'direct' algorithm that generates a quantity equivalent to At , together with such of its derivatives as are required, (in section 4) the iteration process, by means of which successive x. (estimates of x ) are computed; fourth (in section 5) the starting formula (or formulae) for provision of x0 i fifth (in section 6) iteration (when sixth (in section 7) the formulae for computing the is complete'), and the accuracy obtained as a result; 4N velocity components; and the basis for the cessation of and 0 ; third 'convergence , r1 , r 2 from x lastly (in section 8) the rationale behind, and results of, the testing of the solution procedure. 2 CHOICE OF PARAMETER FOR THE ITERATION VARIABLE 2.1 Lambert's theorem and the relations of L-congruence and L-similarity Foa the iteration variable, x , it is desirable to use a quantity that is a 'Lambert 2.2), we require a preliminary digression on Lambert's theorem; as a result of invariant' of the problem, To explain this (in section if possible. ., - *i 4 this theorem, and the equivalence classification of triangles that it makes possible, individual Lambert problems can be divided into equivalence classes.

I Iii Lambert's theorem is usually stated, with an extension of the notation c if is , then At the chord (aide (for a connecting orbital path) can be expressed as a u ), Vi:. already introduced, as fa ltows: CPI'P 2 (multivalued) function of just three quantities (tint counting ri + r 2 negative for hyperbolic orbits 327); many text-books2-4 prove the theorem for elliptic orbits, and a general Sarnecki 'minimalist' proof has recently been given by as the semi-perimeter of the triangle, so that , this last being the semi-major axis of the path (taken as PIP 2 ) of the triangle . Defining c and 28 a s r$ + r 2 = 2s - c motion per unit mass, we can also express At , and noting that a as a function of s , c and is equivalent to the total energy of the energy. It follows from the theorem that triangles with the same values of s and c are equivalent, from the viewpoint of the relation between At and the set of all triangles GPIP 2 can thus be divided into equivalence classes, and energy, r 2 , and the general class (with 0 < c < v, as illustrated in Fig 1. Each class contains a unique (apart from orientItion) isosceles triangle with r, - illustrated in Fig Ia) contains a pair of degenerate triangles such that one of the points PI for either of these degenerate triangles is necessarily rectilinear. Classes With c = 0 (illustrated in Fig 1b) contain only a single member each, which simultaneously isosceles and doubly degenerate, is The other extreme (illustrated lies between the other point and C ; a connecting orbit and P2 in Fig 0c), occurring 'hen c a s , is such that the classes have their widest ratios possible, in regard to the r2 :rI membership, degenerate; each class contains a pair of doubly degenerate triangles such that either PI or P2 coincides with C , whilst the remaining (singly degenerate) triangles (infinite in number, as in the general case) all have P2 on opposite sides of C . Connecting orbits for the singly degenerate triangles of this extreme case cannot be rectilinear, on the usual assumption that C though all members are now is a and P point of reflexion (at infinite velocity) for rectilinear orbits; however, a connecting orbit for either of the doubly degenerate triangles is bound to be rectilinear (as with singly degenerate triangles in the general case). extreme classes with c = s 'degenerate' as referring only to the doubly degenerate triangles. If, we cease to distinguish between the pair of degenerate triangles with , then we can say, and r1 > r 2 exactly one degenerate triangle, is convenient to regard the term in all cases, , therefore, it that an equivalence class contains further, r I < r2 For the mO

Triangles in one of the foregoing equivalence classes may be described as 7 'Lambert congruent', or L-congruent for brevity, and introduction of the concept of congruence suggests the allied one of similarity, just as in elementary geometry. two triangles may be described as L-similar whenever they have Thus the same value of c/s , this being a dimensionless quantity. Though we continue, in section 2.2, to introduce Lambert invariance on the basis of b-congruence, in section 3 we shall find that thinking in terms of L-similarity, with its wider equivalence classes, has the effect of reducing (by one) in the algorithm for the flight time. the number of arguments 2.2 Lambert-invariant parameters, Lambert-equivalont problems, and the parameter x s it Since At , c and energy, is a function only of follows that the equivalence of L-congruent triangles provides the basis for a classification of individual Lambert problems into their own equivalence classes, each such class being defined by the underlying class of triangles and the given value of At . Then a Lambert-invariant parameter may be defined as one that has the same value for all members of an equivalence class of problems. though negates the virtues of the unrestricted angle that were noted in section 1. We is not, which at first sight are Lambert-invariant, is unfortunate that, and It e s c get the best of both worlds, however, by taking 0 (instead of 0) as a parameter of the general triangle, where 0 is defined as being the equivalent isosceles triangle: then 0 can be regarded (like 6) as an angle of unrestricted magnitude. JO , denoted by (The quantity , was recognized 0 for the f as an important parameter in the paper19 by Battin, Fill and Shepperd.) The energy-equivalent orbital parameter a is certainly Lambert-invariant, but this is not true of e (eccentricity) or p (semi-latus rectum). The use intuitively appealing, because of its direct p as iteration variable is of relation to true anomaly and hence to 0 , and it recent as Ref 24. But The paradox is p iterate at all in these circumstances: because At fails, in a somewhat paradoxical fashion, when 0 w p is given, as 2r r 2/(r1 + r,) , without the need to is not involved in this recommended in a paper as that is formula, however, no further progress can then be made without iterating on some other variable. The advantage in using a Lambert-invariant parameter as the iteration that its determination ; ariable is the individual problems of an equivalence class. The resulting 'reduction cases' is a very practical consideration for the solution procedure to be is a numerically identical procedure for all in * Scan .10

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On the Solution of Lambert's Orbital Boundary-Value Problem.pdf

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