Pergamoml 0094-5765(95)00121-2

Acra Astronaurica Vol. 38, No. 1, pp. I-14, 1996 Copyright 0 1996 Ekvier Science Ltd

Printed in Great Britain. All rights reserved 0094-5765/96 515.00 + 0.00

OPTIMAL LOW-THRUST RENDEZVOUS USING EQUINOCTIAL ORBIT ELEMENTS?

JEAN ALBERT KECHICHIANS The Aerospace Corporation, P.O. Box 92957, Los Angeles, CA 90009-2951, U.S.A.

(Received 25 February 1994; revised version received 8 February 1995)

Abstract-The problem of optimal low-thrust rendezvous using continuous constant acceleration is presented based on the use of the non-singular equinoctial orbit elements. State and adjoint equations are integrat’ed numerically by applying the thrust vector along a direction that maximizes the variational Hamiltonian at each instant of time. The two-point-boundary-value problem is solved via a New- ton-Raphson scheme after guessing the initial values of the Lagrange multipliers as well as the total flight time. The transversality condition for the Hamiltonian at the final time is also used to carry out the 7 x 7 search.

Equinoctial elements have previously been used to carry out minimum-time orbit transfers by considering only the five slowly varying elements. The inclusion of the sixth element representing the fast variable has allowed us to extend the applicability of the non-singular formulation to problems of orbital rendezvous. Copyright 0 1996 Elsevier Science Ltd

1. INTRODUCITON

Analytic solutions for the low-thrust rendezvous problem in the vicinity of the initial circular or elliptic orbit have been obtained in the early sixties by Gobetz and Edelbaum[l,2]. In [3] Marec and Vinh have presented both analytic and numerical tech- niques to solve the more general low-thrust problem with rigorous treatment of the analysis of the conju- gate point.

In this paper the problem of minimum-time low- thrust orbit rendezvous using continuous constant acceleration is presented. The variation of parameter equations are written in terms of a set of non-singular equinoctial orbit elements which are free of singular- ities for zero eccentricity and zero and 90” inclination. This orbit theory was developed early in the 1970s by Broucke and Cefola and later extended by Edelbaum and his colleagues to solve five-state orbit transfer problems[4-71. The full set of six-state adjoint and state equations using the mean longitude as the fast variable is used in this paper to solve the optimal rendezvous problem. It is assumed that there exists enough control authority to prevent the use of coast- ing arcs and that the relative geometry between the two orbits is such that a direct rendezvous trajectory is possible. This Ican be achieved by waiting in the

tPaper IAF-92- 14 presented at the 43rd Inrernational Astro - nautical congress, Washington D.C., U.S.A., 28 August-5 September 1992. This work was supported by the U.S. Air Force Space Systems Division under Con- tract No. F-04701-88-C-0089.

IAssociate Fellow AIAA.

initial orbit until the proper orbital configuration is reached for direct trajectory rendezvous initiation. If coasting arcs are allowed, the optimal initial waiting period will be obtained from the optimization pro- cess. Nevertheless, in our simpler problem, the thrust is on continuously from time zero, and as an example we consider a rendezvous problem in near-circular condition to show the effectiveness of using the equinoctial elements as opposed to using the classical elements which would undoubtedly introduce undesirable singularities during the numeri- cal integration of the various exact equations of motion.

2. THE DIFFERENTIAL EQUATIONS IN TERMS OF THE EQUINOCTIAL ELEMENTS

The basic set of the six-state differential equations

of motion have been developed in [4,6,7] and are repeated in this section for completeness.

The equinoctial elements in terms of the classical elements are given by

a=a

h = e sin(w + f2)

k =ecos(w +a)

p = tan (‘>

f sinf2

(1)

(2)

(3)

(4)

q=tan (‘>

i cosR

I=h4+w+i2 (6)

2 Jean Albert Kechichian

The last expression that defines the mean longitude can be replaced by

F=E+w+R (7)

Or

.L=e*+w+i2 (8)

defining the eccentric longitude and true longitude, respectively. Here O* represents the true anomaly. The inverse transformation is given by

a=a

e = (h* +k2)‘12

i = 2 tan-Q2 + q2)‘/*

(9)

(10)

(11)

actan-’ ! 0 0 = tan-‘(E) - tYi(,,

(12)

(13)

M=1--tan-’ k 0

. (14)

If F or L are used as the fast variable defining the sixth equinoctial element instead of 1, then the last expression in eqn (14) should be replaced by

or

E=F-tan-’ k 0

(15)

O*=L--tan-’ k 0

. (16)

The direct equinoctial frame is shown in Fig. 1 with unit vectors ?, & ti, such that ? and 8 are contained

Fig. 1. Equinoctial frame and thrust geometry.

in the instantaneous orbit plane with the direction of 1 obtained through a clockwise rotation of an angle R from the direction of the ascending node. The position vector and velocity i are such that

r=XJ+ Y,d (17)

i=B,3+ F,;,g (18)

where the components are written in terms of the equinoctial elements and the eccentric longitude F, with s, and c, standing for sin F and cos F, etc...

X,=a[(l -h2jI)c,+hkBsF-k] (1%

Y,=a[hkpc,+(l -k2j?)sF-h] (20)

& = a2nrm1[hkj?c,-- (1 - h2j?)sF] (21)

f, = a2nr-‘[(l - k2&-hkps,] (22)

with /I given by eqn (23) and where n = p ‘i2a-3/2 represents the orbit mean motion; p is the earth gravity constant

and where

1

B=(l+G) (23)

G = (1 -h* - k2)‘12.

Here r = a(1 -kc,- hs,). Letting z = (a, h, k,p, q, A)T represent the orbit state vector at time t, and letting f and m stand for the thrust vector and spacecraft mass, respectively, with f =fil where H is a unit vector in the direction of the thrust, the variation of parameters equations for the thrust perturbation are given by eqn (24) with the addition of an n term to 1

where f, = f/m represents the instantaneous pertur- bation acceleration, with both azjai and H expressed in the direct equinoctial frame and where

=M,,f+M,,&?+M,,lk (25)

-h(qY, -pX,)n-‘a-‘G-‘tt

=M,,T+M,,g+M,,f9 (27)

aP nm,a--2G-1

%= KY, _, ---f+=h44,?+M,,~+M,,~ 1,

(28)

a4 n-,a--2G-,

G=Kx' c, --i?=A4M,,l+M,2~+M,,~

~=~-,~-2rl,,G~~~+~~~,, (2g)

+n-,a-'[ -2Y,+G(ha~+k+b

+n-‘a-2G-‘(qY, -pX,)f+

=Ms,P+ Me2:g + M,,lk (30)

with K = 1 +p2 +q2 and

Optimal low-thrust rendezvous 3

lL=n+ z 0

T

.fi!L w where n, the mean motion, is oscillating too since it is dependent on the semi-major axis through eqn (35). Since we are integrating the mean longitude 1 as the sixth element and due to the fact that A’,, Y, , 2, , Y, , r as well as the partials ax, /ah through aY, /ak are given in terms of the eccentric longitude F, it is necessary at each integration step to solve Kepler’s equation

1 =F-ks,+hc, (41)

for F, through a Newton-Raphson iteration scheme. The intermediate results for the partial derivatives

of the position vector with respect to the six elements used to generate the partials in eqn (24) are given by

- ; MS - SF) 1 (31)

ax, hkj?’ - -a

ak- (hc,-ks+ ++ 1

+ F sF(sF - hb) 1 (32)

hk/!l’ -- (hc,-ks+ +) 1

+ f cAkB - CP) 1 (33)

~=.[,,,-ks,,(,.$$)

+ f s&- kj?) 1 . (34)

The system of di:fferential equations for the state variables is given by

. aaT

a= a, 0 . f!! (35)

(36)

~=X,~+Y,~=~[q(Y,T-x*,O)_X,l)] ap dr -=x1$+ Y,~=;,,(x,,- y,f)+ Y,Q] a4

3. THE EULER-LAGRANGE DIFFERENTIAL EQUATIONS

The Euler-Lagrange or adjoint differential equations for the set a, h, k, p, q and 1 have been derived in Refs [8] and [9] and are shown in this section for the purpose of the discussion. The necess- ary condition for optimality, which consists of maxi- mizing the Hamiltonian of the system, can be found in Refs [lo-121.

The Hamiltonian of the system is given by

H = ITi = AT M(z, FxV;B -t I,n (42)

where the seventh differential equation for the mass flow rate has been neglected. The Euler-Lagrange equations are then given by

(37)

ap T P= 5

0 ’ wi (38)

. 84’ 4= a, 0 ’ f!! (3%

The optimal thrust direction II is chosen such that it is at all times parallel to IzM(z, FX in order to maximize the Hamiltonian. The matrix M for the set (a, h, k,p, q, A) is as given by eqns (25)-(30). The partials aM/dz appeared in Ref. [8] and are also

4 Jean Albert Kechichian

shown here in the Appendix. If we let u,, us, u, represent the components of Q in the equinoctial frame, then

H=[~,(M,,~,+M,~u~+M,)u”)

+&M,u,+ M,,u, + M,,u,)

+&W,,++ MS,+ M,,u,)

+iG%,u,+ W+p+ M,,u,.)

+ &WV u, + M,, up + M,, u, 1

+&W,,u,+ KG+ + &u,M + i,n. (44)

The Euler-Lagrange equations are now written as

-1, ( dM,, -,,~,+~..) da oa

where

dn 3n -=--= aa 2~

_;Pl!‘a-‘.’

>

-1; aM,, a&, a&, -- ah u/+ ah ug+-jp (46)

&_dH_ k- dk - [ ( _a dMll

o ak

aM,, aM,, - - ak ‘J+ ak %+xU”

_i aMe. aM,, aMa -i dk (-.‘+,ug+k”’ (47)

aM,, dM,, - - u,+ aP u,+Fu,.

( II

-ah *uJ+- dM,, aM23 ZP

ap % + 7 uw >

aM,, u,+dquw

aK.3 ug + dqu”. (49)

J = _dH= _. A

[ ( WIU +ahu +a% - _ a.4 ‘0 aa J aa g -Z-u” >

aM2l aMI -- al ‘f+ aa_ %+T”” >

-II,,- - ( aM,, aM3, ah433 aa ‘f+ al 3+,,‘*

>

( II

a42 a43 -5 OM”u,+diug+p

aa . > _I

aM,, aM5, -5 $+,+7gg+7pw ( >

_a aw, awl a43 i -g++d).ug+-pw ( >If I (50)

Optimal low-thrust rendezvous 5

The following partials are used to generate the dM/az partials of the Appendix. Here F must be considered to be independent of a but not of h, k and 1 because aA+ is now equal to zero.

ar r -_ =- aLl a

ar a2 -. = 7 (h - SF) ah ar a2 --y(k-c,) ak al $ = a&s, - hcF)

aF 7%; =o

aI: a

ah;= -TCF

aF a -.=_s ak lF

aF a -. =_* a;c r

It is also true that alZ/aF = r/a. The components of the unit vector Q are obtained from

* _ (nTM)T -m’ (51)

From Fig. 1, the thrust vector T = f =jil is defined by the thrust pitch and thrust yaw angles 6, and 6,,, respectively, in the rotating P, t?, 6 frame; P is a unit vector along the instantaneous position vector r, with din the orbit plane and along the direction of motion, and 6 along the angular momentum vector. Therefore

6,~ tan-’ !!! 0

(52) U0

6!,= tan-’ t 0

(53)

with u,, u, and u,, representing the components of the unit vector fl along the P, 8 and 6 directions. We have

f=$+$ (54)

Yl 4 8=6xP=-T;jl+T;jfi (55)

such that

Xl Yl Icr =lU,+Tu* (56)

where

u. I= Yl 4

-,u/+-u, r

u, = u,

r = ]r] = (X: + Y:)tn

(57)

(58)

4. NUMERICAL RESULTS

Since we are minimizing the total rendezvous time, i.e.

s

If J= dt = (tr- to) (5%

this is equivalent to maximizing

J=l;Ldf :_r rr

dt = -(tr- to). 10

(60)

Therefore L = - 1 such that the transversality con- dition at the unknown final time tr namely Hf = 0 for the augmented Hamiltonian H = - 1 + ITi is equivalent to J-& = 1 for our Hamiltonian H = ifi. The two-point boundary value problem consists of guessing the initial values of the six Lagrange multi-

pliers (G, @&,, (G, (&&, tQO, (&), as well as the rendezvous time tf (we assume here t, = 0) and integrating numerically the 12 differential equations of motion namely eqns (35)-(40) and eqns (45)-(50) using the optimal control given in eqn (51) such that the six state parameters at time tf namely (a),, (h),, (Ic)~, (p),, (q)r, (A),, and H, = 1 are satisfied. This consists therefore of a 7 x 7 search with given initial

parameters (a),, (Jt)0, (k)O, (P&, (4&, (A),. For this purpose, we make use of the minimization

algorithm UNCMIN of Ref. [13] which is designed for the unconstrained minimization of a real-valued function F(x) of n variables denoted by the vector x. This subroutine is based on a genera1 descent method and uses a quasi-Newton algorithm. In the Newton method, the step p is computed from the solution of a set of n linear equations known as the Newton equations

V*F(x)p = -VF(x). (61)

Therefore, the solution is updated by using

xk+l =x&+p

=x, - [V’F(x,)]-‘VF(x,) (62)

Here VF(x) denotes the gradient of F at x while V*F(x), the constant matrix of the second partial derivatives of F at x represents the Hessian matrix.

The Newton direction given by p is guaranteed to be a descent direction only if [v’F]-’ is positive definite, i.e. zT[v2Fj-‘z > 0 for all z # 0, since then for small c,

F(x + a) = F(x) + cVFTp + o(c*)

= F(x) - cVFT[V2F]-‘VF + o(c*) (63)

such that F(x + cp) <F(x). This is equivalent to requiring the linear term in Newton’s quadratic approximation for the function F to be negative. This is, of course, the second term in the Taylor series expansion for F at x +p.

6 Jean Albert Kechichian

The algorithm UNCMIN builds a secant approxi- parameters ar = 42767.073 km, er = 1.64459 x lo-‘, mation Bk to the Hessian as the function is being if = 28.343”, Q = 29.999”, or = 247.299” and h4r = minimized such that at x, 120.905”.

Bkp = -VF, (64)

with the matrix Bk positive definite since there is no guarantee that [V2F]-’ will always be positive definite for each x. Once the descent direction is established, a line search on a is established such that F(xk + up) c Fk and the solution updated via X, + 1 = x, + up. The approximate Hessian Bk is updated next using xk+ , and the gradient VF(x,+ I). For example, if F is quadratic [ 131,

B~+,(x~+I-xt)=VFk+,-VFk. (65)

Only gradient values are needed for the update of the approximate Hessian, which is achieved by finite differencing such that the user must provide only the function F itself.

The function to be minimized is chosen as

F=la-%l -++h -h,l+Ik -krI+lP-Prl

aT

+Iq-qTl+y+jff-lI (66)

The value of F is minimized at F = 0.38989 x lo-’ with the solution given as (,I,), = 7347.174908 s/km, (,I,), = - 6.994059338 x lo5 s, (A,), = 8.246922768 x 105 s, (a,),= -3.631161518 x logs, (Qll = - 6.198272749 x 10’ s, (,I,), = - 1.107002405 x 106 s/rad, and tr = 86402.453 s. Figure 2 depicts the evolution of the semi-major axis and eccentricity during the 24-h transfer, while Fig. 3 shows the evolution of the inclination and the right ascension of the ascending node. The relative minima in the R curve correspond to the stationary points of the i curve with the maxima corresponding to the largest negative gradients of the inclination. Figure 4 shows w and M, which exhibit large gradients as soon as the eccentric- ity reaches the near-circular condition. Figure 5 shows the time history of the Hamiltonian which is a constant in this formulation due to the indepen- dence of the accessory variable namely the eccentric longitude on the semi-major axis. Figure 6 shows the control angles 8, and oh that achieve the desired orbit transfer. Finally, Figs 7-9 show the evolution of the Lagrange multipliers during the transfer.

where all the values are at the final unknown time rr, and where the subscript T is used for the target parameters that must be matched. These are, of

course, the same as (a),, (h),, (k),, (P)r, (q)r and (A),. Starting from a, = 42000 km, e,, = 0, i,, = 28.5”, Q, = 30”, o, = lo”, M,,= 0”, and using a constant acceleration f, = 3.5 x lo-’ km/s2, we seek the mini- mum time solution that achieves the final target

5. CONCLUDING REMARKS

The minimum-time low-thrust orbit rendezvous using a set of non-singular elements is presented. This theory can be extended in order to allow the appear- ance of initial and intermediate time coasting arcs to solve minimum-fuel time-fixed rendezvous problems

42.8

42.7

42.8

Z E’ 42.5 Y

” 42.4 .v, 8

.& 42.3

42.0

-1 \ \ \ \

0.003

0.002

8 12 16 20 Y4

Time, hours

Fig. 2. Evolution of semi-major axis and eccentricity.

optimal low-thrust rtndavous

29.99

8 29.98 Ci

s

29.97

180

160

140

B 120 i 2;

I 100 00

Ii 60

Time,hours

Fig. 3. Evolution of orbit inclination and node.

-

-

-

-

28.60

28.48

28.46

28.44 # .,'

28.42 g

P 28.40 3

28.38

28.36

28.34

40

0

280

240

0 4 8 12 16 20 24

lIme,hcwn

Fig. 4. Evolution of mean anomaly and argument of perigee.

M 1/l-_)

Jean Albert Kcchichii

1.8

1.6

1.4

I, 1.2

f 1.0

3 0.8

0.6

0.4

0.2

0

6

1

0 4 8 12 16 20 24

lime,hours

Fig. 5. Evolution of optimal Hamiltonian during transfer.

I I I I I

0.33

0.32

B 0.31

s ai- d

0.30

$ 0.29

8 Ti 0.28

e 0.27

0.26

40

30

2o 8 -0

.

10 a= .

0 P

8 -10 g

2 -20 c

-30

-40 4 8 12 16 20 24

Time,houra

Fig. 6. Optimal thrust pitch and yaw control angles during transfer.

Optimal low-thrust rendezvous 9

-0.6 r

-1.6 - t

1.8

E 1.6 f

E .

1.4 +

1.2 H E m

1; fI&,,,/V I . 0.6

0 4 8 12 16 20 24

-363.10

-363.14

g -363.18 .g 5--

p63.22

it -363.26

-363.30

Tme,hours

Fig. 7. Time history of A, and &.

-619.80

-619.85

-619.90

*19.95 2 ,o

-620.00 E _

-620.05 <

it 420.10 x

420.15 j

-620.20

-620.25

_ y620.30 a 12 16 20 24

Tme,hours

4

Fig. 8. Time history of lp and A,.

10

4 7.25

3 7.20 z 5 7.15 J+ q 7.10

B 7.05

7.00

6.95

Jean Albert Kechichian

6.90 I ‘9’ I I ’ 0 4 a 12 16 20

-0.96

-0.98 i

-1.00 Fi

-1.02 : x

-1.04 5 T

-1.06 g

-1.06 3

Time.hours

Fig. 9. Time history of 1. and I,.

as well. Further improvements would necessitate the consideration of the J2 zonal harmonic and possibly the acceleration due to air drag for low-earth orbit applications. Although the present work presents exact solutions using precision integration, long dur- ation rendezvous applications can effectively make use of the averaging technique for rapid calculation.

1.

2.

3.

4.

5.

6.

I.

8.

9.

10.

REFERENCES

F. W. Gobetz, A linear theory of optimum low-thrust rendezvous trajectories. JAS XII, 69-76 (1965). T. N. Edelbaum, Optimum low-thrust rendezvous and station keeping. AZAA J. 2, No. 7 (1964). J.-P. Marec and N. X. Vinh, Optimum low-thrust limited-power transfers between arbitrary elliptical orbits. Acta Astronautica 4, 51 l-540 (1977). R. A. Broucke and P. J. Cefola, On the equinoctial orbit elements. Celestial Mech. 5, 303-310 (1972). P. Cefola, Equinoctial orbit elements: application to artificial satelite orbits. AIAA 72-937, AZAAIAAS Astrodynamics Conference, Palo Alto, CA, 11-12 September (1972). T. N. Edelbaum, L. L. Sackett and H. L. Malchow, Optimal low thrust geocentric transfer. AIAA 73-1074, AZAA 10th Electric Propulsion Conference, Lake Tahoe, NV, 31 October-2 November (1973). P. Cefola, A. Long and G. Holloway, The long-term prediction of artificial satellite orbits. AIAA 74-170, AZAA 12th Aerospace Sciences Meeting, Washington, DC, 30 January-l February (1974). J. Kechichian, Equinoctial orbit elements: application to optimal transfer problems. AIAA 90-2976, AZAA/ AAS Astrodynamics Conference, Portland, OR, 20-22 August (1990). J. Kechichian, Trajectory optimization with a modified set of equinoctial orbit elements. AAS/AIAA 91-524, Astrodynamics Specialist Conference, Durango, CO, 19-22 August (1991). A. E. Bryson and Y.-C. Ho, Applied Optimal Control. Ginn and Company, Waltham, MA (1969).

11. J.-P. Marx, Optimal Space Trajectories. Elsevier, Amsterdam (1979).

12. N. X. Vinh, Optimal Trajectories in Atmospheric Flight. Elsevier, Amsterdam (198 1).

13. D. Kahaner, C. Moler and S. Nash, Numerical Metho& and Software. Prentice-Hall, Englewood Cliffs. NJ (1989).

APPENDIX

The Partials of the M Matrix

The partial derivatives of M with respect to h

aM,, 2 a& ah nza ah

(Al)

aM,* _ 2 a& ah n2a ah

aMI -=O ah

642)

643)

(A41

In a similar way

aM22 -=

ah na2(1 _;L’_,,,,*(~ - y 9

(‘45)

a‘% -= ah

hW -~~;W3”~qy,_pX,)

+k(qZ -pig (A6) na*(l -h* - k2)“*

Optimal low-thrust rendezvous 11

aw, -=

ah

ah (i+p2+q2) ar, hi, -=

ah 2na2(1 -h2-k2)‘/2 [ ah+(l -h2--kq 1 (AlO) 86 PC

ah I . (All)

The partials

aM,, a.v,, aM5, m2 -=__=_ ah

=- =O dh ah ah

are all identically zero.

6412)

(A13)

aM6, (l-h2-k2),-1/2 -= -

+ h(1 - h2- kz)-‘(qY, -pX,) 1 . (A14)

The partial derivatives of M with respect to k

i&u,, 2 aY, -- =__ ak n2a ak

(Al%

i&u,, 2 af, _-a-- ak n2a ak

6416)

dM,3 -0

dk- (A17)

aM2, bly, -PX,) 1 -=

ak

na2(1 _ h2 _ k2)‘/2 + na2(1 - h2 _ @l/2

fc(q$ -p$!)+;y;;y;))] (A20)

a&, -= ak

k (“+kj+) na2(1 - h2 - k2)‘j2 ah

(1 -k2 _kZ)V -

.,f!i;~j+~)$+;~] (A21)

a42 -=

ak ,,$(I _;2_k2),,2r$+kB;)

6422)

a43 -= dk

-na2(l _~_k2)3,2(qYI-p~~) (~23)

aMu (1 +p2+ 42) ar, -= ak 2na2(1 - h2 - k2)‘/* dk

k(1 +p2+q2)

+ 2na2(1 - h2 - k2)‘j2 Y, (~24)

a&, (I +p2+q2) ax, -= ak 2na2(1 - h2 - k2),j2 ak

k(1 +pz+q2) x (A25) +2na2(l _h2-k2)312 ’

a‘% _ 8M.32 a&, m2 =-=--.-=o

dk ak ak dk

(A261

12 Jean Albert Ke-chichian

aM, (1 -h2-k*)-“2

ak= na2 [(q$ -PZ)

+k(l -h2-k2)-‘(qY,-pX,) I

(A28)

The partial derivatives of M with respect to p

The only non-zero partials are

aM2, -kX, ap= na2(1 - h2 - k2)“* (A29

w, h& ap= na2(1 - h2 - k2)1’2 (A30)

8% PY, ap= na2(l - h2 - k2)‘12 (A31)

aM5, PX, p= aP na2(1 - h2 - k2)12 (~32)

aM5, -Xl -= aP na2(l _/,2_k2)‘/2 ’ (A33)

The partial derivatives of M with respect to q

The non-zero partials are

aM2, W -=

a4 na2(1 - h2 - k2)‘j2 (A34)

8% -hY, 7= na2(1 - h2 - k2)‘j2 (A35)

aMu qY, ag- na2(I - h* - k*)“* (A36)

a% qX1 -= 84 na2(1 - h2 - k2)“2 (A37)

a&., Yl -= a4 na2(1 - h2 - kz)‘” ’ (A38)

The partial derivatives of 2, with respect to h and k are

2 =$%, [

sF+;cF(ksF-hcp) 1 +$

* { hPsp+&cr+hs,)(~ +s)

+; c,[hkjs,+ (1 - h2B)cF] (A39

a$ ’ -= -:a ak

-cF+fsF(ksF-hcF) 1 +g r

* f$ (kc, + hs,) + hpc, - f s,[hkjs,

The partials of f, with respect to h and k are

al: 8 2

-=--a ah r

-s+(ksF-hcF) +“a 1 r

. $ (kcF + hs,) - k/?s,

+[hk/?c,+(I (A41)

(~42)

The second partials of X, and Y, with respect to h and k are

(A43)

a2x, 24 W' y@= -~sF~+8)+cCF--ksF)

a2x, -= ah ak

W4

(A45)

~=a~s,(j?+&)-(hc,-ks,)

- sF)

(AW

NJ) ’ -f(h -s,)+s, 1 6447)

Optimal low-thrust rendezvous 13

Next, the accessory partials a2X, /aa ak, a2X, pa ah, a2 Y, /aa ak and a2 Y, /#?a ah, are generated with aF/aa = 0. In all the following partials, 8X,/& = X,/a and aY, / aa = Y,/a

a*x, i ax, daak a ak

(A51)

a%, I ax, -=__ aaah a ah

(~52)

av, i ay, -=__ aaak a ak

(A53)

av, i au, -=--. aaah a ah

(A54)

The partial derivatives of M with respect to a

(A55)

(A56)

aM,, ---CO aa

(A57)

aM2, (1 _ h2 _ k2)1'2

-=

aa na2

(A58)

aM22 (1 _ h2 _ k2),'2

-= -

aa .a2

--!-!-;+f&!$-;~ 1 (A59)

aM2, k

-z-= na2(l - h2 -- k3,/2

1 (A@3

ah (1 _ h2 _ k2)“2

XL=- na2

.--!$+!%+gX,+~!Y$

[ 'I

6461)

w2 (1 _ h2 _ k2),/2

-=_

aa na2

. ++!s+g~,+~~ [ .I

(A621

a% -h -=

aa na2(l - h2 - k2),/*

.[-;(qY,-*X,)+$&P2 (A63) 1 WI ---GO aa (‘464 m2 ---CO

aa a%, (1 +p2+!12) -=

aa 2na2(l - h2 - k’),” (-LY,+Z) ::I

aM,, --.---CO aa

(A67)

aM,, -=O aa

(A68)

aM,, (1 +p*+43

aa= 2na2(l - h2 - k2),‘2 ( -&x, +z

> (A69)

a%, -M~I 1 ==7+t;;;I

. -2$!+(~-h~-k9V~. [

hbz +kjs >I (A70)

aMe2 -=

aa

-[-22+(1-h2-k2),“. >I (A71)

a43 -=

aa -!5+~

.[(j!&p!$‘)(, _h’-k2)-“2-j (Aj’2)

with

s_ aa

- -$hk&-(1 -h2/3)sF] 6473)

$=f;.[hk&-(1-k’/T)c,] 6474)

The partial derivatives of M with respect to 1

aM,, 2 ait, an n2r aF

6475)

aM,2 2 af, -=-- aA n2r aF 6476)

aM,, -.-..-=O a2 (A77)

aM2, (1 - h2 - k2)“2 -= aA nar (

d2X, hf? a& ---- aFak n aF >

(A78)

a&l (1 -h2-k2)“2 a2Y, hja?, -= ---- a2 nar aFak n aF 6479)

Jean Albert Kechichian

aM2, k(q$ -Pz) -= aA nar(1 - hZ - k2),‘*

a&, (1 -h* -k*)“* 8*X, k/I aA+, -=-

an nar aFah+naF >

aw2 (1 -h* - k*)“* a2Y, k/3 a$ -=-

ai MT aFah+naF >

34, a&2 --=-=o an an

8% (I +p*+43 ay, -= aA Znar(1 -h* - k*),‘* aF

3% a%2 =. -=- al an

a&, (1 +p* + 42) ax, -= - an Znar(1 -h* - k*),‘* aF

a& 1 -=_ aA nar [

-$+( _h2_/$)‘/*

. h/!lg+k/?g ( >I

aw.2 1 ay, -=-

a3, -2-+((1_j,*_k*)‘/*

nar aF

>I

(4% -42) aM,,

an - nar(1 - h2 - k*)“*

(AW

6481)

6482)

(A831

(A841

(AW

(AW

(A87)

(AW

(A89)

(A901

The auxiliary partials are

ax, aF = a[hkbc,- (1 - h*j?)sJ (A911

w aF = a[-&?~,+ (1 - k*fl)cJ (~92)

a*, - = -9 (ksF- hc&, + $f [-hkjs,- (1 -h*&] at+

(A931

a$ -= -;(ks~-hcf)~,+f+tk/?c,-(l -k*j?)sJ aF

(A94)

g =a[(hs,+kc+ +s)

+ ; (h/? - sF)(sF- h) + f c$ I

(A95)

a*x, -= aFak

--a [

-(hs,+ kc,\hkS’ l-8

a* +7(sF-h~)(cp-h)+%pcF 1 (A96) r

g=, -(hs,+kc,)ff [ 1-B

-$(k/3-c,)(s,-h)+fs,c, 1 (A971

~=‘[_,hs”+kc++~)

+;(c,-kfi)(c,-k)-;s; . I (A981