Acta Astronautica Vol. 9, No. 3, pp. 139-146, 1982 Printed in Great Britain.
0094-5765/82]030139-~$03.00/0 Pergamon Press Ltd.
OPTIMAL IMPULSIVE TRANSFER WITH TIME CONSTRAINT KARL G. ECKEL'[" 26222 Escala Dr., Mission Viejo, CA 92691, U.S.A. (Received 3 Februnty 1981; in revised [orm 27 August 1981)
Abstract--The primer vector theory developed by D. F. Lawden is applied in order to solve the problem of optimal orbital transfer by impulsive thrust. The condition of a minimal characteristic velocity is constrained by the condition of a fixed transfer time or by the rendezvous condition. The development is performed for noncoplanar, elliptical orbits and for an arbitrary number of impulses. The solution is given by a set of m equations, where m equals the degree of freedom of the minimum problem. The transition to the solution of the free time problem may be performed in the final result. 1. INTRODUCTION
By the assumption of impulsive thrust the transfer trajectory consists of a sequence of orbital arcs, which intersect each other in points called junction points. Here the thrust is applied, by which a velocity increment is added to the path velocity of the preceding orbital arc. The sum is the path velocity of the following arc. The magnitudes of the velocity increments, which are denoted by V1 (i = 1, 2. . . . . n), are called velocity impulses. The sum V=~
V~
is called characteristic velocity of the transfer trajectory. Optimal orbital transfer is defined by minimal propellant expenditure. This minimum is obtained by minimizing the characteristic velocity. By the optimal performance treated here a constraint concerning the transfer time is added to the condition V = minimum. The characteristic velocity and the transfer time are functions of a finite number of variables. The standard method to solve such minimum problems is given by the theory of extrema, by which the solution is obtained in the form of m equations called optimum conditions, where m is the degree of freedom of the minimum problem. It has been shown in[2] and[3] that the extensive work of differentiating, which is required by application of the theory of extrema, may be avoided by application of the primer theory derived by D. F. Lawden in[l]. The solution obtained in this way for the most general form of the free time problem consisted as well of m optimum conditions. The same method will be applied here in order to solve the time constrained minimum problem. The equations given in[l] for the components of the primer and its time derivative contain an integration constant C, which is zero for the free time problem. These equations have been applied in[2] and[3] by setting C = 0. Here they are applied in the complete form. tResearch Specialist with Lockheed Aircraft Corp., retired. eThe nomenclature is given in the appendix.
However, the solution is valid as well for the free time problem, since the condition C ---0 may be introduced in the final results, where the constant C is preserved as a parameter. 2. THE APPLICATION OF THE PRIMER THEORY
The concept of the primer vector arises by the application of the calculus of variations to optimal transfer problems as performed in[l]. For the coasting parts of any transfer trajectory the components of the primer and its time derivative are determined by given equations as functions of the true anomaly along each orbital arc. In general, the primer and its time derivative vary discontinuously at junction points. However, for optimal transfer trajectories the following statements hold true: (1) The primer p and its time derivative [~ are continuous along the whole transfer trajectory. (2) At junction points, the primer and its time derivative are orthogonal, from which it follows that p[~ = 0. (3) At junction points, the primer is a unit vector in direction of optimal thrust. (4) If the minimum condition is constrained by the condition of a fixed transfer time defined by the difference of the terminal times tt and to, the previous statements hold true, provided that all junction points are interior points of the time domaine to < t < t,. These qualities together with the given equations for the components of the primer and its time derivative will be used in order to determine the optimum conditions. The components A, /~, v of the primer and the components ~, 7, ~ of its time derivative are defined in a Cartesian coordinate frame, A and ~ in radial direction, tt and "0 horizontal and in the orbital plane, and v and perpendicular to this plane. These components are given for elliptical orbits by eqs (5.55)-(5.57) and (5.63)-(5.65) of[l] as functions of the true anomaly f. By taking over these equations we change the denotations of the components into A', t~', v', ~', "0' and ~' and we replace the symbols 11, /2 and I3 into I, J and K respectively. Then we substitute the dimensionless magnitudes x, sr, ~/and ( defined by the polar equation.~: r l x. . . . (1) I l + e cos[ 139
140
K. G. ECKEL
and by the equations:
g]k = -- AE sin/~k + BMx,k + X,k(Dk -- Ak sin f,k) + Cd~k, (7)
where ,/ is the gravitational constant. In this way Lawden's equations become ;,' = A cos f + Be sin / + CI,
(8)
~k = AE sin ilk -- D~ - (Bk - C~K,k)lxi~,,
(9)
"O'~= X,k[--AE(e~ + COS[,~) + C~ cos .f~ + Dkek sin [,k], (10)
/~' = - A sin .f + B/x + x(D - A sin/) + CJ, v' = x(E cos / + F sin/),
~ = X,k(EK COS/,k + F~ sin/,k),
(3)
= A sin / - D - (B - CK)/x,
~k = X,k[-- Ek Sin.f~k + F~(e~ + cos /,~)],
(11)
( i = 1,2 . . . . . n;k = i - 1, i).
= x [ - A ( e + cos f) + C cos / + De sin/], = x[-E sin/+F(e+cos/)].
Herein the symbols I, J and K are defined by I = sin
/f
d/
sin2/(1 + e cos/)2,
J =x2c°sf+l xe sin / '
K=x(xsin/-J)
(4) (5)
The symbols A, B, C, D, E and F denote constants of integration. They are constant, as well as the eccentricity e and the semi-latus rectum l, along each orbital arc, but differ on different arcs. Equations (3) will be applied to all junction points of the transfer trajectory. At these points all variables and parameters take on some values, which are specified by subscripts, with subscript i (i = 1, 2 . . . . . n) referring to junction points, and subscript k referring to an adjacent orbital arc. The subscript k takes on either the value i - 1 referring to the arc that precedes the junction point P, or the value i, referring to the following arc, as shown in Fig. 1. Magnitudes which refer to junction points and arcs have two subscripts, the first one referring to the junction point. By these denotations eqs (3), applied to all junction points, become X~ = A~ cos f,k + B~ek sin/,k + C~I~,
Y
(6)
r,+l
Fig. 1. Consecutive orbital arcs. Notation: C center of attraction, Pi, Pi+~ junction points, V~, V~+~velocity impulses, n, n+~ distances from center of attraction. Subscripts refer to junction points: 0~-~,0i, 0~+~orbital arcs, A.6 transfer angle. Subscripts refer to orbital arcs. They equal the subscript of the previous junction point..f,, .6+~.~true anomalies. The first subscript refers to a junction point, the second to an orbital arc.
The values of the variables and parameters f~k, x~k and ek contained in these equations are determined for any given transfer procedure by the general transfer equations. As well the direction cosines ~,~,#~k and v,k of the velocity impulses and the characteristic velocity may be computed by these equations. They are listed in/3/for n impulses. The optimum conditions are obtained by formulating the special qualities which the primer and its time derivative take on in optimum solutions. Since, in this case, the primer is a unit vector in direction of the thrust, the components A', 0.' and v' of the primer become the direction cosines X, 0. and z, of the thrust. Therefore, the following relations are valid for the optimum solution.
p.~.i-i = P.i.i-h ~.~
~ = V~,~-~,
/~]i = ~il, ~
(12)
= V~
The condition of continuity of the primer and its time derivative requires that at junction points p~.~-i= p, and p~.H=p,. This condition is already satisfied for the primer by eqs (12), since the thrust direction is uniquely determined by the general transfer equations. The components ~.~ ~ and ~, and the components ~,~.~-1and u, must be distinguished only, because they are determined in different coordinate frames. The condition of the continuity of the time derivative requires first that at all junction points ~.~-1 = ~. This relation may be brought into the form: /~-~_~,.,_, = ~,,, (i = 1,2 . . . . . n)
(13)
by application of the first eqn (2). The correspondent equations for the other components, which can be equalized only in connection with a rotational transformation, have been established in/3/, in order to derive the optimum conditions for the free time problem. They would be sufficient here as well to solve the constrained minimum problem, but are replaced by the condition of orthogonality ;t~k + t~krl~k+ ~k~Ik = 0 in order to simplify the development. If herein eqns (2)
Optimal impulsivetransfer with time constraint and (12) are inserted, this condition takes the form Xi~ik+/zi~'0i~+ vl~li~= 0 (i = 1, 2. . . . . n; k = i - 1, i).
141
after the direction cosines ;,i~, ~ik, and v,k have been substituted by means of eqns (12). For simplicity eqns (7) may be replaced by the equations:
(14)
~k + ~ik = (C~ - AE)XiE sin .fie + DK(Xi~-- 1), Equations (12) may be applied as well to eqns (6) through (8), which are valid then for the components A~k, ~ik and V~k. More relations are needed, which are not explicitly given by the primer theory. They are obtained by making use of the Hamiltonian integral which is represented in[I] by eqn (5.39). The right hand member, which may be denoted by H ' is constant along the whole optimal transfer trajectory and is zero for free time transfer. If the symbol H ' is preserved in the subsequent development up to eqn (5.55) without absorbing other constants, it is found that between the constants C and H' the relation c=lZH e3,
,
which result by adding eqns (7) and (9) and inserting the second eqn (5). By the method of substitution, first the constants B~ and then CE and D~ may be eliminated. By a frequent application of the Pythagorean identity and of the polar equation the following explicit equations are obtained: C k e k : l ( X ' + ~ Xik ik+~lke~sin.fik)
Akek = - - Xlk
p~ikek sin .flk + "Oik,
(17)
(18)
Bkek = l (\Xik Kik + t.~ikXikCOS.fie -- ~ikl,k sin fik - "OikAk), (19)
exists. Since l and e differ for each orbit, we may write D~ek = ~l[Ai sin .flk - I.~ikXik(ek+ COS[ik) Xlk
ci=li2H. ei
(i = O, 1,2 . . . . . n).
(15) -
~:ik COS fik + rllk(Xik +
1) s i n .fik],
(20)
where H = H'/% The constants C~ and H are zero for the optimal free time solution. Equations (6)-(11) and eqns (13) through (15) are 16n + 1 equations, which must be satisfied by the optimum solution.
Ek = Vik(ek + COS.fik)-- lie sin l,k,
(21)
FE = ViE sin [ik + llJ, COSfik
(22)
3. THE REDUCTION OF THE NUMBER OF OPTIMUM CONDITIONS
The product Akek for a fixed subscript k = s (s # 0, s # n) is contained in two different equations of the set (18), namely for i = s , k = i and for i = s + l , k = i - l . By equalizing both expressions for A,es the product A~e~ is eliminated. In the same way all products Akek, Bkek, Dkek and the constants EE and FE may be eliminated except for k = 0 and k = n. By this elimination process the following equations are derived from eqns (18)-(22).
The choice of any free time transfer procedure has 4 n - 5 degrees of freedom. For instance, the true anomalies of the points of departure and arrival, one parameter of each of the n - I transfer orbits, and the three space coordinates of each of the n - 2 junction points which do not lie on the terminal orbits may freely be chosen. Therefore, the degree of freedom of the constrained minimum problem is 4 n - 6 , and as many optimum conditions are needed in order to solve the optimum problem. However, the 16n + I equations mentioned before contain 6n +7 constants of integration, namely A, B. C, D, E, E (i = 0, I, 2. . . . . n), and H, and 6n magnitudes ~k, ~ik, and liE, which are not determined by the general transfer equations. Therefore, 12n +7 equations are needed to determine these unknown values. The remaining 4 n - 6 equations correspond to the number of freedom of the minimum problem. Explicit equations may be established for the unknown values mentioned before. However, these values are not needed to represent the optimal performance. Therefore, we shall establish the explicit equations only to the extent necessary to eliminate these values from the remaining 4 n - 6 equations, which, in this way, become the proper optimum conditions. The explicit equations for the constants of integration are obtained by solving eqns (6)-(11) for these values,
(i= 1,2 . . . . . n ; k = i - 1, i).
)~ - tz,ei sin fi~ + "O, = xji ~ - t~ieisin fjl + ~jl, x,
1 [),-i
Ki, +
(23)
,,xii cos I . -
l ( X , K,i + tz,ix,i cos fii - ~,ilv - "0vJ,i ], / Xji \ Xji
(24)
& sin/,i - ~,(ei + cos .fii)- ~ cos fii + xii + 1 sin fi~ xl, Xii Xi--~--T~ii sin .fj, - m~(ei + cos fj,) - 6 i cos .fj~ + xj~ + 1 , sin h , xii xj--~- r/r
(25) ~,ii(ei+ cos [.) - ~, sin .f. = v~j(ei + cos hi) - Cji sin hi, (26) v,, sin I, +lii cos .fil = vii sin hi + lJi cos hl ( i = 1,2 . . . . . n - 1;./= i + 1).
(27)
142
K. G. ECKEL
The constants of integration with the subscripts k = 0 and k = n can not be eliminated, but the correspondent eqns (18)-(22) are discarded from the optimum conditions, since they serve only to calculate these constants, after the optimum problem is solved. The constants Ck, which are contained in eqns (15) and (17), can not be eliminated in the same way, but will be preserved as parameters. Equations (24), the only ones containing the magnitudes I, J and K, will be simplified by means of the parameters Ct. After inserting eqns (5) for J and K, eqns (24) contain the magnitudes Idsin[i~ and I~dsin~ with different coefficients. However, a comparison with eqns (17) shows that both coefficients equal the same constant Ct. Therefore, eqns (24) may be brought into the form (Xt + ~l,t)cot [ , , - (X~+ rl~t)cot ]~t = e~(C,Y,- R,),
In order to eliminate the magnitudes ~,, ~t, 0,, O~t, ~tt, and ~v implicit equations will be established by solving eqns (17), (23), and (26)-(28) for these magnitudes. Substituting the polar equation for x, and xv in eqns (23) and using the parameters S~ defined by the equations: St=A, cosf~t-ht c o s f , - I ~ v sinf~t+lzt, sinf,t. (36) eqns (23) become
~, + ~ , , - ( ~ + 0~,) = etSt.
Solving eqns (28) and (37) by Cramer's rule for Xt + 0, and ~j + Ov yields: e, s i n
(28)
(37)
X, + 0,, = ~
f,t.-.
Ira',r, sin fj, - Rt sin fj, - S, cos f~t), (38)
where _ e, sin6, . ~ . '~ + 0~, - ~ ~tS,r, sin f, - Rt sin f,, - S, cos [~,).
R, = X~s i n h t - X, sinf,, + ~, cosf~,- m, cosf,t,
Y, = I~dsin[j, - hdsin.ftt.
(29)
(39)
(30)
From the definition of R~ and Si given by eqns (29) and (36) it is readily found that
It follows from eqn (4) that Y~ is the definite integral
R, sin [ji + St cos/'j, = Xj - At cos A[, - / ~ , sin a.f~, f t~' df Y' -- J~,, sin2[(l + e cos [)2.
(31)
In [4] the indefinite integral is given for e < 1 in the form
f d[ tan(.¢/2) cotq/2) J sin f(l + e cos [) = 2(1- e) - 2(1+ e)
(32)
Introducing the eccentric anomaly ~ defined by , = 2tan
_,
l-e [ ( X / ( l - ~ e ) tan ~),
sin e
X/(1 - e ~) sin .f 1 + e cos.f ' (33)
and eliminating the half angles by means of known trigonometric formulae the definite integral may be brought into the form y, =
Inserting these relations into eqns (38) and (39) and applying for simplicity the parameters X~ and Xv defined by X, = ?'j - '~ cos A.f~ OL, x,, sin M, x,'
6e 2 -t 1- e ( 1 - e - ~ tan ( ~ / ( l - - ~ e ) tan ~) e 3 sin [ + (1 - e2)2(1 + e cos [) + constant.
Rt sin f, + St cos f,, = ~,i cos Af~ - ~, - pj, sin Aft.
(1 + e,2) sin Afi - 2el(sin [j, - sin [.) (1 - et2)2 sin f, sin fv
the wanted implicit equations for 0i~ and "OJ~become A, + 0,,
', x, sin Af,
X,, eix,, sin ft,,
(41)
Xj + OJ,
= (G~sinf, \ xjt sin Af,
) Xjt e,xj, sin fj,.
(42)
(i= 1,2 . . . . . n - l ) Equations (17) written once for k = i and once for k = i - I and solved for ~t.l-t and ~it become Chxih
~h = sin.f,h _ 3e~2%1 - ~ . ) - e~3( s i n ~v - s i n E,t)
Xj-~= Xj cos A f i - X~ V-J, xjt sin AI, xj,' (4O)
Ai + "rlth xtheh sin.fd
Cixu
~" = sin [,,
Ai + 0, x,e, sin [d
(34)
(! - 62) ';2
(43) (i-- 1.2 . . . . . n ; h - - - i - l )
where Inserting eqns (41) and (42) yields Mt = .6, - [,,,
(35) / x.
(See Fig. 1.)
Yi sin lit ~ + X
'~"=C'~.~Tm-.I. xi, si.,X¢.,/
"
143
Optimal impulsive transfer with time constraint 6~ = C i ( ~ v
x~lY~sinf'l~'X~sin Aid *
(44)
Herein we insert the relations sin fj, sin Afi= cos f, - cos fu cos Afi,
(i = 1,2 . . . . . n - 1) sin/, sin Afi = -cos/ji + cos/, cos Afi. Hereby eqns (43) for ~Lo and ~ . have not been applied. They are discarded from the proper optimum conditions, since they serve only to determine ~t.o and ~.. after the problem is solved, Solving eqns (26) and (27) for ~,~ and ~i yields
Then we factor out cos A/~ and apply the polar equation. In this way the expression (49) becomes - •, c o s A f,
x, \
sin A!~
/~li)- I ( A j cos A f t - A ~ - p , ) sin Afi
v~- vii Afi ~" = x~ sin Af~+ v. tan--~-,
+ (Xi + Xj) tan - ~ .
v~- vii Af~ = v~ tan -~-. £~ x, sin Af~
(45)
Equations (41), (42), (44), and (45) are valid for i-1,2 . . . . . n - 1 . Equations (25) will be used in order to determine the parameters G. We eliminate the magnitudes 6~, and 6~ for convenience by means of eqns (43). This leads to the equation A~+ r/,
~
-
C~ = ei
X~i - Xj~ + (Xi + X~)tan(AH2) c o t h , - c o t . f , + Y , L ~2ei ( s , n . h , - sin fii) + 1+ ei2]
r
(50) (i= 1,2 . . . . . n - l ) .
[2ei + (1 + e,~) cos .fll]
X~+ ~/~i .~ + (1 + e~~) cos fi,] e~ sin f~ lzei
= C, (cot f,i - cot/ji ) - Ri - e, (m, - g,)
(46)
Herein R~ is determined by eqn (29). By application of the relations cos Afi cos f. = cos hl + sin f,, sin Af. cos Afi COS fie = cos f . - sin hi sin A f,. which are derived by expanding cos A[, and applying the Pythagorean identity, it is found that A~- A~cos sin Afi Afi cos f,
This expression is the right hand member of eqn (48). By use of the parameters X,~ and Xj, eqns (48) solved for C~ become
Aj cos Afi - h~ sin Af~ cos h~ = A~ s i n f~, -
Ai sin f . .
The explicit expressions for the magnitudes g, 0, ~ and C have been derived from eqns (6)-(11) given for the primer and its time derivative. The optimum conditions are obtained by substituting these expressions in the remaining equations which represent the conditions for the optimum, namely eqns (13)-(15). The magnitudes with the subscripts o and n referring to the terminal orbits can not be eliminated, but the equations which contain these magnitudes are discarded from the proper optimum conditions. Together with the previously discarted equations they determine the magnitudes in question but are not needed to solve the minimum problem. Only one of the constants Ci, say C., is needed to determine the Hamiltonian constant H. It follows from eqns (15) that es
(47)
The elimination of the sums a~ + "O. and ~ + ~ from eqns (46) by means of eqns (41) and (42) and the application of eqns (47) yields
H = ~ C.. 1, Then eqns (15) in the form ei
~C,-H=O e~.i[ yi(2e, sin f6,l ,-sAfi isin n
(i= 1,2 . . . . . s - l , s + l
..... n-l)
÷ 1 +e,2)+cotfi,-cotf,]
(52)
1 - cos AI~ = e~(~q sin fji-A~ sin f , ) + ~-~J~-W~+ 2(X~+A~) x~ x. sin Afi (48)
are n - 2 optimum conditions. Elimination of ~, and 6~ from eqns (13) by means of eqns (44) yields immediately
• / l • [ _ / xih
The right hand member of this equation may be transformed into [Aj(ei sin fv sin Af~ + 1 - cos AfO - h~(e, sin f. sin Af~ -l+c°sAli)]~
(51)
1 + "gjl~ v - ttii+ Afi ~ (A,+Aj)tan--~-.
Yh sin fhh '~ + X~h]
(~, = Ci
~
Y, sin f,, "~ xil sin All/+
X.
(53)
( i = 2 , 3 . . . . . n - l;h = i - 1,j= i+ 1) (49)
The elimination of ~?, "0 and ~ from eqns (14) by means
K. G. ECKEL.
144
of eqns (41), (42), (44) and (45) leads after a small rearrangement to the following equations. Xii
CiYi sinful\ (~, - ~ sin S T ) ( A , - ~,,x,,e, sin f,,)- ~, + ~',,( Pjl~- -
(Xj~ C~Y~s i n / , \
- xj~ sin Af~ )Oq -
bfii
+
Af, \ v,, tan -~-) = 0,
Cix~i
sin Af,,/ (54)
sinL~)-Xi(~j,- C~xj~ sin AL,]
t~j~xj~e,
validity of the optimum conditions. Since the difference between the terminal time and the time of the impulse may be arbitrarely small, a solution which requires that To or T, is zero must be accepted. However, it is not permitted to enforce such a solution by setting To = 0 or T, = 0 in the constraint (58). The flight times T~ must be established as functions of the true anomalies of the junction points. For simplicity we write the flight times as functions of the eccentric anomalies E in the form
/Jii - - /~ii
T, = ~ (a~-y)[Ej, - e,, - e,(sin ej, - sin e,,)] (i= 1,2 . . . . . n - l )
(i = 0, 1,2 . . . . . n),
Equations (52)-(55) are the desired 4 n - 6 optimum conditions. For n = 2 the sets of eqns (52) and (53) are empty and the set (50) contains only one equation, which serves to determine the parameter Ct. Each set of eqns (54) and (55) contains one optimum condition. In order to solve the free time problem, by which the degree of freedom is increased by one degree, the condition C~ = 0 is introduced in all equations, which in this way are essentially simplified. By setting C~ = 0 in eqns (50) we get the n - 1 optimum conditions X,, - Xi, + (~ + ~,j)tan(Afd2) = 0 (i = 1,2 . . . . . n - 1), (56) which replace the n - 2 optimum conditions (52). These optimum conditions for the free time problem have already been derived in [3].
4. T H E
CONSTRAINT
The optimum conditions with C~ 0 are valid for a constraint which requires that the transfer is performed during the time period to
T = tl -
to =
~_~ Ti,
(57)
i=o
where T~ are the flight times on the orbital arcs. We assume that the value of to is predetermined by its definition as the earliest time where the spacecraft is ready to start into the transfer orbit. Therefore, we define the constraint by the equation ~"~ Ti = T*,
(58)
i--I
where T* is a fixed value of the transfer time determined by the choice of the terminal time t~ alone. The solution of the transfer problem has to satisfy the optimum conditions and the constraint besides the general transfer equations. The condition of no coincidence of terminal point and junction point refers to the
(59)
where a~ are the semi major axes of the orbital ellipses. The true anomalies may be substituted herein by means of eqns (33). The expressions obtained in this way for T~ must be substituted in the constraint (58). The eccentric anomalies Eoo and e,+l., do not refer to junction points, but are fixed parameters. The value of coo is determined by the position of the spacecraft at t = to. With respect to the fixing of ~.+t., and T* the problems are different. If these values may freely be chosen, the following relations must be considered. Two pairs of terminal values, (f,+t.,, T*) and (f, +L,, T*), satisfy the same constraint, if T* = T* + A T, where AT is the flight time between the true anomalies f,+,., and [.+k,. Since the optimum conditions do not contain the angle [,+t.,, the optimum solution is the same for both pairs of values. A spacecraft that moves on the target orbit determines varying pairs of values (f., T) for each point of the target orbit. If these pairs are considered chosen values of terminal points, they all have the same optimum solution. The optimal transfer trajectories differ only by the length of the arc of the target orbit. This length becomes zero, if the true anomaly f,, of the solution equals the chosen value f.+L.. If the solution is obtained for f.n >f.+~.., the flight time T. is negative. However, such a theoretical solution may be transformed into a realistic solution by increasing f.+~., sumciently and adding the correspondent flight time to the transfer time. Because of this possibility to transform the solution, it is not necessary to choose f.~ 1.. very large in order to assure that the condition f . ~ . . > f . , is satisfied by the solution immediately. If the transfer is performed in order to observe an event from a certain point of the target orbit at a certain time, the values of f.+t,, and T* both are predetermined by the space mission. Therefore, this problem has only one solution. If this solution is obtained for f.. >/.+1.., it can not be transformed into a valid solution. In this case the problem can only be solved by an additional constraint. By the rendezvous problem the position of the target at the time to must be given by the true anomaly .to.. The flight time T, of the target from its position at t = to to a chosen terminal point with the true anomaly f,+t., is
Optimal impulsive transfer with time constraint predetermined by the equation ~tt 3
T, = X/(~-)[,~+l.n - ,o. - en(sin ,~+~.n - sin ,on)], (60) which corresponds to eqns (59). Rendezvous is achieved, if the flight times of spacecraft and target from their initial positions to the terminal point are equal. Therefore, the predetermined flight time Tt may be used as the fixed time T* in the constraint (58). This yields the rendezvous condition in the form:
•
T , = T,.
i~O
(61)
The flight time T, may be separated in two parts, namely the flight time T', from the initial position to the point of rendezvous with the true anomaly .f,n and the flight time T,, which both spaceships fly together. Thus 7", = T', + 7",, where T', is determined by T~=~/(~)[~,,-~o.-(sin~,,-sin~on)l,
(62)
If the flight time Tn, given by eqn (59) with i = n, is subtracted from both members of eqn (61), the rendezvous condition becomes n--I
Z 7",= T,, i=O
(63)
where in the left hand member the expressions of eqns (59), and in the right hand member the expression of eqn (62) must be substituted. 5. C O N C L U S I O N
The optimum conditions derived here from certain results of the primer theory represent the general solution of the most important problems of optimal transfer by impulsive thrust. The conditions must be satisfied together with the general transfer equations in order to determine the optimum solution. A certain problem is established, if values are given for the parameters which define the configuration of the terminal orbits. For special orbit configurations the optimum conditions of the free time problem have been simplified in [2], [3] and [5], but explicit solutions can not be derived. For the coplanar, coaxial configuration, where explicit solutions exist, the optimum conditions fail, since they become indeterminate expressions. However, the optimum conditions are simple tools to derive numerical solutions by means of the NewtonRaphson method. This way to obtain numerical results for an optimum problem, which is called indirect method, has the advantage of the fast convergence of the Newton-Raphson method. The ratio between the correction and the previous deviation increases in each step of the approximation, approaching the value one. By the method of gradients, which is usefully applied as long as no optimum conditions have been derived, this ratio
145
varies in each step, but does not increase in the average. Consequently, more steps are needed in order to obtain the result with the same accuracy. The unfavorable judgement about the work load of the indirect method given in ([6], Chap. 1.51), considers only the work load in each step, but, even in this limited area, it is not applicable for the optimum conditions derived here. The simplicity of these optimum conditions, compared to the general transfer equations, is not immediately recognized. The work load of the computer is determined mainly by the number of values which must be computed by a series expansion, since they need a multiple of the running time needed for a multiplication. There are 23 such values in the general transfer equations of the two impulse mode, but none in the optimum conditions of the free time problem. Hereby it is assumed that values which are used repeatedly are computed only once. Based on these numbers it may be deduced that the application of all three optimum conditions adds only a few percent to the work load needed to apply 15 general transfer equations. However, the general transfer equations are applied by the indirect method only four times, and by the method of gradients at least six times in each step. The advantage of the indirect method is even more evident by constrained minimum problems, since the method of gradients needs an essential modification, while the basic procedure of the indirect method remains unchanged. The constraint contains 12 values to be computed by a series expansion, the optimum condition contains none. The apparent complexity of the optimum conditions consists only in summations and multiplications. REI~ERENCES
1. D. F. Lawden, Optimal Trajectories for Space Navigation. Butterworths, London (1%3). 2. K. Eckel, Optimum transfer between non-coplanar elliptical orbits. Astronautica Acta 8, 177-192(1%2). 3. K. Eckel, Optimum Transfer in a Central Force Field with n Impulses. Astronautica Acta 9, 303--324(1%3). 4. D. F. Lawden, Fundamentals of space navigation. J. Br. lnterplan. Sot?. 13, 87-101 (1954). 5. K. Eckel, Numerical solutions of non-coaxial optimum transfer problems. J. Astronautical Sciences 10, 82-92 (1%3). 6. G. Leitmann, Optimization Techniques. Academic Press, New York (1%3).
APPENDIX
Nomenclature
e f I n P r T~ T V~ V x = r/I 3' Af e
eccentricity true anomaly semilatusrectum number of impulses junction point distance from center of attraction flighttime on orbital arc transfer time velocityimpulse characteristic velocity dimensionless distance parameter gravitational constant anglebetween two consecutive junction points eccentricanomaly
146
K.G. ECKEL
A, ~, v direction cosines of thrust, A in radial direction, horizontal in the orbital plane, and v perpendicular to this plane The subscript i (i = 1, 2. . . . . n) on P, r, V and A refers to a junction point. The subscript i (i = 0, 1, 2. . . . . n) on e, I, T and Af refers to an
orbital arc. The arc preceding the junction point P~ is denoted by the subscript i - 1 , and the following arc is denoted by i. The magnitudes ,f, x, ~, ~ and v, which refer to a junction point and to the preceding or following arc, have two subscripts, the first one referring to the junction point, the second one to the arc.