INDIRECT OPTIMIZATION METHOD FOR IMPULSIVE TRANSFERS

Abs t r ac t

Guido Colasurdo* and Dario ~as t rone t

Politecnico di Torino Torino - Italia

In this paper a method for obtaining minimum- fuel finite-thrust trajectories is extended to impulsive manoeuvres. The method follows the classical formu- lation of the problem which is provided by the theory of optimal control. Some characteristics of the solver for the resulting boundary value problem and a pecu- liar enforcement of the switch conditions guarantee a very accurate and fast numerical solution. Impulsive burns are considered as points where some state and adjoint variables are discontinuous. Examples are pre- sented in order to show that a wide variety of orbital manoeuvres can be treated by means o i a unique tech- nique which only requires a low performance personal computer.

I n t roduc t ion

A great number of papers have dealt with the problem of optimizing orbital transfers; therefore no general reference will be given here. This vast interest was initially due to the large benefit in terms of pay- load, deriving from strategies which make clever use of the propellant. Orbital transfer has consequently become a classical bench mark for ~pt~imization pro- cedures, giving a reason for the persistent presence in literature of papers which deal with problems whose practical solution is already well known.

The authors' interest in theoretical aspects of op- timization procedures addresses their attention to- wards indirect methods, even though direct methods permit an easier treatment (often the treatment) of the most complex problems. At the 1992 and 1993 AIAA/AAS Astrodynamics Conferences, the authors presented an indirect method for optimizing finite- thrust non-coplanar time-open' or time-fixed2 orbit transfers. Attent,ion has now been devoted to impul- sive manoeuvres. Relevant literature could suggest to an inattentive reader that the optimization of finite- thrust and implilsive transfers requires different indi- rect methods. The aim of this work is to show that

'Professor, Dipartimento di Energetica, Member AIAAIAAS t Researcher, Dipartimento di Energetica, Member AIAA

impulsive manoeuvres constitute only a limiting case of the more general finite-thrust problem and can be solved by means of the same procedure, for instance the quite general technique which has been developed by the authors.

In this paper the optimal control problem (OCP) is first formulated for the finite-thrust manoeuvres in order to describe those peculiarities of the authors' ap- proach which enhance the efficiency of their indirect procedure. The capabilities of their solver for the re- sulting boundary value problem (BVP) are highlighted for the same purpose. The straightforward extension to impulsive manoeuvres is therefore presented. The method is first applied to time-open coplanar manoeu- vres using a canonical trasformation which avoids any integration of differential equations and reduces the problem to the iterative solution of a set of algebraic equations. Different iterative procedure^^-^ have re- cently been suggested in order to optimize the impul- sive transfer between coplanar elliptical orbits. The present procedure appears to be more interesting, be- cause it retains all the features of the indirect ap- proach to more complex problems and can therefore be irnmediately extended to non-coplanar and/or time- fixed problems, while the other aforementioned pro- cedures are confined to coplanar time-open problems. Time-fixed coplanar and non-coplanar examples are in fact presented. In these cases, and in time-open non- coplanar problems which are not here given as exam- ples, the analytical treatment of the coast arcs would be, a t least, complicate: the numerical integration of the differential equations is a priori selected.

In the last example a non-coplanar rendezvous is executed by means of a finite-thrust burn followed by a coast arc and by a final impulsive burn in order to prove that the same procedure can contemporaneously manage both thrust levels.

BVP solver

The indirect approach usually transforms an OCP into a BVP which must be numerically solved. The efficiency and capabilities of the optimization proce- dure are strongly related to the capabilities of the nu-

Copyright 0 1 9 9 4 by the American Institute of Aero- nautics and Astronautics, Inc. All rigths reserved. 441

1st bum j coasting i 2nd bum

0 1 2 3

time-like variable E

Fig. 1 - Sketch of a two-burn transfer.

merical method which solves the BVP. As the OCP formulation has to match the solver characterist,ics, a preliminary description of the BVP solution method seems to be opportune.

The BVP solver is a ~ubst~ant ia l improvement of the solver which has been prodlicecl by Gasparo et According to the Newton method, their code reduces the BVP to a snccession of initial value problems, which are solved by means of an Adams-type integra- tor, with variable step-size and order of accuracy, as suggested by Shampine and Gordon6. The integ~at~ion interval must be prefixed. The original code permits linear constraints on the values which the variables assume a t the two e x h i o r points (or external bound- aries).

The improved solver can split the integration in- terval into e suhint~ervals, by adding (e - 1) znter- nu1 boundarzes. As only fixed boundaries are permit- ted, without any loss of generality the boundaries are located (Fig. 1) a t the consecutive integer values of the independent variable E on the interval [0, el. More general (even non-linear) boundary conditions are ac- cepted and each subinterval may be ruled by distinct equations.

The mathematical f~ rmula t~ ion of the problem, as previously outlined, consist,^ of e systems of n first or- der equations:

which must be successively integrated from & = 0 to

E = e l with 12 boundary conditions:

11" (xn (0), xn ( l ) , . . . , x n ( j ) , . . . , xn (e)) = 0 (2)

If the vector sm, where m = n(e + I ) , collects the val- ues assumed by the dependent variables a t the (e + 1) boundaries, and if (n-k) boundary conditions give ex- plicit initial values to ( n - k) corresponding variables, Eqs. (2) can be rewritten in the form:

Eqs. (1) are integrated by using, as initial values, Eqs. (4) and an estimation (pk)r of the remaining k variables:

After the first estimation (pk)o , the integration is re- peated from the improved initial values provided by the relation:

until Eqs. (3) are satisfied. The matrix in the above equation is evaluated as the product of two matrices:

where the first only depends on t,he structure of Eqs. (3)) while the second is made up of the values taken a t the boundaries by tthe matrix:

and is obtained by consecutively integrating the e sys- tems of differential equations:

which, by making the Jacobian of Eqs. (1) explicit, is written in t,he more useful homogeneous form:

T h e initial matrix gnrk(0), according to its definition, is composed of an identity matrix and a zero ma- trix ~ ~ - ~ l ~ .

The independent variable is frequently left open the statement of practical problems, but, by means a simple normalization, can be transformed into the

variable E. The differential system, which now depends even on q unknown constant parameters bQ, takes the form:

which can be reduced, from the conceptual point of view, into the form of Eqs. ( I ) , by adding q further differential equations:

and by gathering the vectors yn-Q and bQ into the vector x n . From the practical point of view, the code however integrates the ( n - q) Eqs. (11) and then the significant k(n - q) Eqs. (10). During each iteration, (k + l ) ( n - q) differential equations must therefore be integrated; about ten iterations are required in order to achieve a ten digit satisfaction of the boundary con- ditions.

Another capability of the BVP solver is relevant to the aim of this paper: in the j-th subinterval Eqs. (1) can be s ~ b s t i t u t ~ e d by algebraic relations:

xn ( j ) = q' (xn ( j - 1)) (13)

In the same subinterval even Eqs. (10) are substituted by algebraic relations which are directly deduced from the definition expressed by Eq. (8):

A means to manage dis~ont~inuities in the vector xn is thus provided.

F i n i t e - t h r u s t t r a n s f e r s

When fuel-optimum strategies for finite-thrust transfers are considered, it is well known that relevant trajectories are composed of maximum-thrust arcs

connected by coast arcs. According to the above state- ment, and to the solver characteristics, the present ap- proach regards the patth as being composed of e suc- ceeding arcs or phases. Under a central inverse-square gravitational field, if the initial radius, the correspond- ing circular velocity and the spacecraft initial mass are assumed as length, velocity and mass units, the equa- tions of the motion, during the j-th phase, are:

which must be int,egrated from t j- l to t j , with constant, thrust level (zero during the free-flight arcs). The effective exhaust velocity c is also constant. The aim is to minimize the fuel used, that is to maximize the final mass me.

The Harniltonian is now constructed as:

where the co-state or adjoint vectors are obtained by integrating the Euler-Lagrange eqnations:

The process is autonomous (i.e. not explicitly depen- dent on time) and Eq. (18) is constant over each arc, but the const,ancy is extended to the whole trajec- tory, if neither discontinuities in the state variables nor path ~ons t~ra in t s are considered: the junct,ions of con- secutive arcs correspond to the corners of the theory of optimal control, where the Weierstrass-Erdmann conditions7 prescribe the continuity of the Hamilto- nian and adjoint variables.

The optimal t,hrnst direction is obtained by ap- plying Pont,ryaginls Maximum Principle to Eq. (18):

when singular arcs are excluded, maximum thrust must be exerted parallel to the adjoint vector to the velocity A", termed primer vector by Lawdens.

Eqs. (15)+(17) and (19)+(21) are accompanied by an equal number of ext,ernal boundary conditions which derive from the constraints for the states and the transversality conditions for the adjoints.

Time is now normalized by means of the arc time- lengths ~ j , by posing for each phase:

in order to locate the arc junctions at consecutive in- teger values of the new independent variable E . The introduction of e unknown parameters r j requires e additional boundary conditions. The continuity of the Hamiltonian is imposed a t the ( e - 1) internal bound- aries. As the state and adjoint variables are continu- ous, it is equivalently formulated as:

The left hand term of Eq. (23), or switch function, is the multiplier in the Hamiltonian of the thrust magni- tude T j , which is discontinuous at the internal bound- aries. When optimal time-open trajectories are in- vestigated, the last condition is the vanishing of the Hamiltonian, which is imposed as an initial boundary condition:

but it is an easy task to prove that Eq. (24) is equivalent to the fulfillment of the switch condition Eq. (23) at the initial boundary. In time-fixed prob- lems Eq. (24) is substituted by the time constraint:

switch function step by step in order to put the control a t either its maximun or minimum value. The present procedure preliminarly assumes the structure of the trajectory (i.e. a prefixed succession of full-thrust and coast arcs, if intermediate-thrust arcs are excluded). The nullity of the switch function a t the junction of consecutive arcs is then imposed as a constraint on the solution. By so doing, the switch from full-thrust to null-thrust (and vice versa) is governed, at least a t the beginning of the iterative process, by the assumed arc time-lengths. The method convergence results to be improved, probably because these values are less diffi- cult to estimate than the initial values of the adjoint variables, which actually rule the sign of the switch function.

Impuls ive transfers

The theory concerning optimal impulsive trans- fers is equally consolidated. Its knowledge is here pre- sumed, even though some results can be intuitively de- duced by means of Pontryagin's Maximum Principle. An approximate numerical solution of impulsive prob- lems can be achieved by assuming finite but very high thrust levels. This practical approach constitutes a re- alistic representation of actual spacecraft manoeuvres: nevertheless such a solution is unsatisfactory from the academic point of view. Retaining the general scheme of the optimization procedure which has been out- lined in previous sections, an impulsive transfer can be viewed as a succession of coast arcs separated by the points where impulsive burns occur: some state and adjoint variables are discontinuous there. For prac- tical reasons, the BVP solver treats these points as special arcs whose time length is undefined.

As the optimal t,hrust direction is parallel to the primer vector Xv, Eqs. (13) describing the discontinu- ity are:

The above reformulat,ion of such a classical prob- lem cannot obviously offer new theoretical achieve- ments. But the authors, in their fairly conventional approach, instead of adapting a BVP solver to the rel- evant OCP, formulate the problem in order to match the capabilities of an efficient solver. For instance, a fairly unconventional use of the switch condition is proposed. In a simple problem presenting a bang-bang control, the classical indirect procedure integrates the state and adjoint differential equations, checking the

Relations specifying the continuity of the other vari- ables have been omitted. Moreover, the mass rn and its adjoint variable A, could be neglected when dealing with a mere impulsive problem and Eqs. (27) and (28) would become superfluous. They are retained for the

condition on the primer magnitude:

phase angle of ellipses we

Fig. 2 - Time-open transfer between coplanar ellipses.

.214 ' I

60.0 60.5 61 .O

phase angle of ellipses we

Fig. 3 - Enlargement of Fig. 2.

sake of both clearness and capability to treat trans- fers which alternately use low-thrust and high-thrust engines. For conciseness reasons, Eqs. (14), which can be easily obtained, are not reported here.

The number of unknown parameters which must be determined by the procedure does not change, as the time-length of a finite-thrust burn ~j is substituted by the magnitude of the impulsive velocity increase A?, but only one Eq. (23) is available for each im- pulsive burn. T h e boundary condition which is lost a t each time-open burn (i.e. every intermediate burn and the initial or final burn during a time-open transfer) is substituted by the stationarity of the switch condition or, in force of the same Eq. (23), by the equivalent

Time-open c o p l a n a r t r a n s f e r s

A time-open coplanar transfer is often composed of two burns and a coast arc. I t could be optimized by means of the proposed procedure and using po- lar coordinates. In Fig. 1 the manoeuvre (actually a generic state or adjoint variable) is schematically de- picted as a function of the solver independent variable E . In the case of impulsive burns, they are described by the algebraic Eqs. (13), i.e. an analytical integration (dotted line) is carried out. If a canonical transfor- mation suggested by Fraeijs de Veubekeg-lo is used, both impulsive burns and coast arcs can be analyti- cally integrated. The problem is therefore reduced to the iterative solution of a set of algebraic equations, which is rapidly achieved by the same BVP solver. This appears almost equivalent t o the procedures sug- gested by Lawden3 and by Broucke and Prado4, who respectively solve 11 or 3 (more complex) algebraic equations. The present technique is however aimed a t more difficult problems and its stretched application to the time-open coplanar case is obviously no more than an academic diversion.

A previous paper1' gives details of the canoni- cal transformation: i t uses the radial and tangen- tial velocity component,^, the reciprocal of the angu- lar momentum and the reduced characteristic velocity w = -c log rn as dependent variables, while the cen- tral angle substitut,es the time as independent variable. The complete set of Eqs. (13) for the coast arc can be found in that paper, while the analogous equations for the impulsive burn and the optimal conditions are given by R i z z ~ ~ ~ , who also discusses some examples.

A single case concerning the optimal transfer be- tween coplanar ellipses is here presented. The semi- major axis of the initial orbit with eccentricity e, = .5 is assumed as the reference length (a, = 1); the ge- ometry of the final ellipsis is defined by a, = 1.5 and e, = 113. The total required AV depends on the rel- ative angular position of the ellipses which is defined by the argument of periapsis w, of the final orbit, as w, = 0. For w, > 60°, the ellipses intersect and a single-burn transfer is possible. In Fig. 2 the total AV for the three-burn or bi-elliptic transfer is also shown. Two families of two-burn solutions are found (coax- ial ellipses provide an intuitive case): the procedure attains the family which is closer to the solution as- sumed to start the iterative process. I t is interesting

to note that either burn has A h < 0 (i.e. opposite to the primer) when the achieved solution is not glob- ally optimal. In this way the omniscient primer warns that for a short range of w, the single-burn transfer is optimal (Fig. 3) and vengefully forces the authors to numerically integrate the adjoint system in order to analyze the time behaviour of the switch function.

Rendezvous manoeuvres

As far as time-fixed transfers are concerned, Lion and Handelsman13 have proposed a method which has been used to obtain optimal solutions among others by Prussing and Chiu14 and by cola surd^'^: the latter's code is however slow and not very efficient. The same examples have been now carried out using the present technique which has proved fast and accurate.

The example in Fig. 4 refers to the optimal ren- dezvous with a target in the same circular orbit hav- ing an initial phase angle A29 = K; the maximum al- lowable is time t , = 4 . 6 ~ . The optimal solution is obviously symmmetric: this procedure is quite flexi- ble as far as boundary conditions are concerned and only half the problem can be analyzed. The four- impulse external solution presented by Prussing and Chiu14 (AV = ,189) is accompanied by an znlernal solution which achieves AV = ,164, if no constraint on the minimnm radins is posed. As both solutions satisfy the necessary optimum conditions, an indirect method cannot discriminate between them and con- verges to the closer solution: the user's intuition is an

Fig. 4 - Four-impulse rendezvous.

5 10 15

allowable time te

Fig. 5 - Multi-impulse rendezvous manoeuvres

indispensable complement to these methods.

In the next example the radius of the target cir- cular orbit is r, = 1.2 (A29 = K). The optimum AV is presented in Fig. 5 as a function of t,he allowable t,ime t,. If this time is scarce, a t,wo-impulse manoeuvre is optimal, bnt a t t , = 2.47 a three-impulse and then (t, = 4.60) a four-impulse manoeuvre result to be op- timal. Some cases are shown in Fig. 6. If the time con- straint becomes less stringent, the number of impulses diminishes and initial and final constings appear in or- der to eventually reach the Hohmann transfer global

Fig. 6 - Mult,i-impulse rendezvons manoeuvres.

optimum, when enough time allows to wait for the cor- rect spacecraft phase angle. At t , = 10.70 the three- burn transfer becomes again optimal; an initial coast- ing, which has never been present, is therefore added ( t , = 13.65). At t , = 15.00 the first impulse van- ishes and a two-impulse solution with an initial coast- ing takes over; from t , = 15.14 the Hohmann transfer is available. This example gives the occasion to note a feature of the present technique, which is also present in Lion and Handelsman's indirect approach13 but is typical of direct methods16. The trajectory structure must be preliminarly assumed: for instance, two thrust arcs connected by a coast arc. The numerical code at- tains the best solution compatible with the assumed configuration. If this configuration is not mandatory, the investigation of the achieved solution in the light of Pontryagin's Maximum Principle gives suggestions on the possibility of improving the manoeuvre. For in- stance, a positive value of the switch function during a coast arc recommends the insertion of a further burn. Even this problem has been treated and discussed by Prussing and Chiu14: the basic difference does not con- cern some details of Fig. 5, but the fact that a much more detailed figure has not required a greater effort.

Non-coplanar examples analyze the rendezvous between a spacecraft which was moving on an equato- rial circular orhit and a target moving on an inclined circular orbit. Longit,udes are measured from the line of nodes. The final orbit in the next example has ra- dius re = 1.2 and inclinat,ion ie = 5'. The initial posi- tion of the chasing spacecraft and the final position of the target are particular: they are in quadrature with the line of nodes. The two-burn manoeuvre, which is in a polar plane, is too far from the optimum and can- not be used as the starting solution for the three-burn

1st bum 1st coast j 2nd bum 2nd coast ; 3rd bum I 0 1 2 3 4 5

time-like variable E

Fig. 7 - Three-impnlse non-coplanar rendezvous.

Fig. 8 - Transfers with impulsive or finite 2nd burn.

problem. Once again experience in orbital mechanics suggests a suitable tentative ~ o l u t i o n ' ~ based on two mere increases of the velocity magnitude and a mid- course simple plane change. The optimization proce- dure partially transfers the plane change to the initial burn in order to charge the mid-course burn with a smaller rotation of a smaller velocity (Fig. 7) . The cost is thus lessened from A V = .I705 to AV = .1626.

The last example compares a finite-thrust ren- dezvous (T = . l , c = 1.5) to a similar manoeuvre whose second burn is impulsive. In the latter case.

-60 0 60 120 180

longitude 6

Fig. 9 - Transfers wit,h impnlsive or finite 2nd burn.

both thrust levels coexist in the same procedure, ac- cording to the aim of the paper. The final circular orbit has re = 3 and ie = 10'; the initial longitudes of the chasing and target spacecrafts are respectively 190 = -64.7' and 29t = 65.7'; the allowable time is t , = 8. Figs. 8 and 9 show the trajectory projections onto the equatorial plane and onto a cylinder normal to the equatorial plane. Even though the impulsive burn uses the propellant a t a greater distance from the attractive body, the final mass (me = .705) is larger than in the case of two finite-thrust burns (me = .660): but the same final time is a less tight constraint for the manoeuvre which impulsively benefits from thrust, i.e. without any time delay.

Conc lus ions

A fast and accurate indirect technique for optimiz- ing finite-thrust transfers has been extended to impul- sive manoeuvres. A very simple enforcement of the optimum conditions characterizes this technique which joins together flexibility and efficiency. Even though growing computer performance favors the even more flexible direct methods, the theoretical background of indirect methods justifies their further development, to which this paper hopefully contributes.

This research has been supported by the Consiglio Nazionale delle Ricerche and by tlhe Ministero delllU- niversit& e della Ricerca Scientifica e Tecnologica.

Refe rences

' Colasurdo, G. , "Optimal Finite-Thrust Space- craft Trajectories", Paper AIAA 92-4510, AIAA/AAS Astrodynamics Conference, Hilton Head, SC, August 10-12, 1992.

Colasurdo, G. and Pastrone, D., "An Indirect Method for the Optimization of Finite-Thrust Non- Coplanar Rendezvous Manoeuvres", Paper AAS 93- 741, AAS/AIAA Astrodynamics Specialist Conferen- ce, Victoria, B.C., August 16-19, 1993.

Lawden, D.F., "Optimal Transfers Between Co- planar Elliptical Orbits", Journal of Guidance, Con- trol and Dynamics, Vol. 1.5, No. 3, 1992, pp. 788-791.

Broucke, R.A. and Prado, A.F.B.A., "Optimal N-Impulse Transfer Between Coplanar Orbits", Pa- per AAS 93-660, AAS/AIAA Astrodynamics Speciali- st Conference, Victforia, B.C., August 16-19, 1993.

Gasparo, M.G., Macconi, M. and Pasquali, A., Ri- soluzione numerica di problemi ai limiti per equazioni differenziali ordinarie mediante problemi ai valori ini- ziali, Pitagora Editrice, Bologna, 1979.

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Lawden, D.F., Optimal Trajectories for Space Na- vigation, Butterworths, London, 1963.

Fraeijs de Veubeke, B., "Canonical Transforma- tions an'd the Thrust-Coast-Thrust Optimal Transfer Problem", Astronautzca Acta, Vol. 11, No. 4, 1965, pp. 271-282.

lo Fraeijs de Veubeke, B., "Optimal Steering and Cutoff-Relight Programs for Orbital Transfers", Astro- nautica A d a , Vol. 12, No. 4, 1966, pp. 323-328.

" Colasurdo, G . and Pastrone, D., "Optimal Or- bit Transfer Using a Canonical Transformation and an Indirect Method", Paper AAS 93-339, AAS/GSFC International Symposium on Space Flight Dynamics, GSFC/NASA, Greenbelt, MD, April 26-30, 1993.

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l3 Lion, P.M. and Handelsman, M., "Primer Vector on Fixed-Time Impulsive Trajectories", A IAA Jour- nal, Vol. 6, No. 1, 1968, pp. 127-132.

l4 Prussing, J .E. and Chiu, J.-H., "Optimal Multiple-Impulse Time-Fixed Rendezvous Between Circular Orbits", Journal of Guidance, Control and Dynamics, Vol. 9, No. 1, 1986, pp. 17-22.

l5 Colasurdo, G. , "Sull'individuazione delle traiet- torie impulsive ottimali per rendez-vous orbitali", IX Congress0 Nazionale AIDAA, Palermo, October 26-29, 1987.

l6 Enright, P.J. and Conway, B.A., "Discrete Appro- ximations to Optimal Trajectories Using Direct Tran- scription and Nonlinear Programming", Journal of Guidance, Control and Dynainics, Vol. 15, No. 4, 1992, pp. 994-1002.

l 7 Bate, R.R., Mueller, D.D. and White, J .E., Fun- damentals of Astrodyn,amics, Dover Publications, New York, 1971, p. 379.