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AAS 09-103

ALGORITHM OF AUTOMATIC DETECTION AND ANA LYSIS OF NON-EVOLUTIONA RY CHANGES IN ORBI TAL MOTION OF GEOCENTRIC OBJECTS
Sergey Kamensky*, Andrey Tuchin , Victor Stepanyants, Kyle T. Alfriend§
The goal of this work is develop ing the methods and algor ithms for automatic detection and analysis of non-evolu tionary changes in orbital motion of geocentric objects caused by man euvers and o ther events (but not by natural perturbations). The task is formu lated for three k inds of non-ev olution ary chang es, namely th e sing le impu lse man euver, two impulses maneu ver and small continuous thrust. The methods and algorithms ar e developed. Examp les of the algorithm work are given .

INTRODUCTI ON Automatic determination and adequate characterization of the man euvers performed by spacecraft in near-Earth orbits is a significant compon ent of the general p roblem o f tracking Earth satellites and catalog maintenance. This paper presents the techniques and algorithms for the detection and analysis of non-evolutionary changes of orbital motion of satellites resulting from maneuvers. The p roblem is formulated for three types of n on-evolutionary change of orbital parameters: one burn man euvers, t wo burn man euvers and lo w thrust man euvers. Identification of the non-evolutionary change of the orbit requires rather good knowledge of the satellite orbital motion, which can be acquired by processing of the measurements obtained by the sensors. This paper assumes that the satellite orbits are already determined before the maneu ver and after. In case the measurements were continuous we could consider only one burn and low thrust maneuvers. The techniques described in this work use a nu mber of classical problems . The paper describes the techniques for solving them wi th adjustments required by the go al task. The problem of the determination of the orbital transfers for given initial and final orbits has an infinite number of solutions. Selection of the most probable scheme of maneuvering is based on the evaluation of the characteristic velocity. Selecti on of one of three (one burn, two burn or low thrust) v ariants of man euvering is bas ed on the following criteria: We assume a one burn transfer wh en the orbits before and after the maneuver intersect and the value of the characteristic velocity does not ex ceed a sp ecified limit.

* §

Design man ager, "Vympel" Corporation, Mo scow , Russia Leading research er, "Vymp el" Corporation, Moscow, Russia Leading research er, "Vymp el" Corporation, Moscow, Russia TEES Distinguished Research Ch air Pro fesso r, Dept. of Aerosp ace Engineering, T exas A&M University , USA

1


The t wo burn of the minimum performed using Lambert problem

transfer is al ways possible. The times of the burns are selected by the condition of the characteristic velocity. The ev aluation of the two burn maneuver can be two techniques: by using the matrices of the partial derivatives or by using the depending on the ch aract er of orbital motion.

The lo w thrust maneuv er is evaluat ed in case we kno w the spacecraft is equipped with a lo w thrust engine. The first section of the report considers the mathematical basis of the algorithms for characterization of the maneuvers. We consider the classical problem of orbit determination given two positions and the time for the transfer bet ween them ­ the Lambert's problem and the problem of evaluation of the required characteristic velocity for coplanar transfers. Th e algorithm for the Lambert's problem is based on the techniques suggested by the Russian scientist Subbotin1,2,3. The geo metric method is used for evaluation of energy supply required for coplanar maneuvers. The second section of the report considers the evaluation of the impulse of the burn when we know the orbital paramet ers before and after the burn. In the case of a one burn maneuver the tracks before and after the burn must intersect in the point where the burn was applied. Thus the analysis of the cause of the change of the orbital motion should include determination of the point of minimu m distance bet ween the tracks before and aft er the burn . The resulting vector of residuals between orbital positions before and after the burn should be compared with the error of position prediction based on orbital parameters before and after the burn. The third section of the repo rt considers the case wh en the parameters of t wo orbits are kno wn and two burns perform the transfer between them. The algorithm for determination of the times of performing the burns and ev aluation of the respective characteristic velocities is described. The fou rth section describes the algorithm for determi nation the time interval for the work of low thrust engine and the generated acceleration provid ing the required orbital transfer.

MATHEMATI CAL BACKGROUND Dev elopment of the algorithms for evaluation of o rbital man euvers requires s everal classical methods. Description of these methods is given by Vallado 4. In particular this book includes a description of the methods of orbit determination for t wo given positions and the time interval for orbital transfer (Lamb ert Problem) and the analysis of coplanar maneuvers . This section of the paper considers these cl assical problems . The algorithm for Lambert's problem is based on the techniques suggested by Subbotin1,2,3. The geo metric method is used for evaluation of energy supply required for coplanar maneuvers. Orbit determination for two given positions of the s pacecraft The evaluation of the two burn transfer bet ween orbits 1 and 2 requires an alysis of the set of transfer orbits with further s election of the orbit providing the minimum of supplied characteristic velocity. Each transfer orbit is determined by the positions in orbits 1 and 2 and the respective times for these positions. In the scope of the non-perturbed motion the problem of orbit determination for t wo positions and the time of the trans fer bet ween them is a classical Lambert p roblem. Theorem (Euler-Lambert) For the two vectors r1, r2 of positions of a mass point in the central gravitational field and the time interval

t for transfer from position r1 to position r2 , the following equation is valid:

2


µ
where

ta

3 2

=

sin

(

sin

)

+2 n

(1)

a µ n

­ ­ ­

semi-major axis of the elliptical transfer orbit, gravitational parameter, the number of revolutions of the transfer orbit

sin

2

2

=

r1 + r2 + r2 r1 4a 0 2 ,

, sin 2

2

2 2

=

r1 + r2 4a 2 .

r2 r1

,
(2)

n n =2 + 2 n, 0 < 2 is the arc passed by the mass point, the sign "­" cor1= does not exceed , and the sign "+" to the case wh en exresponds to the case when n n ceeds . 1 and revolution number. n 2

are true anomalies corresponding to vectors r1, r2 taking into account the

Euler ­ Lambert theorem sho ws that the time for the orbital transfer of the mass point from one position to another depends only on the sum of the radii of these positions r1 + r2 , on the value of the chord connecting them r2

r1 , and on the semi-major axis of the orbit a . Equation

(1) is also called Euler-Lambert's equation. For historical referen ce1 we note that this theorem for t he parabolic orbit was first proved and published by Euler in 17435. Lamb ert's theorem, gen eralizing Euler's result was found in 17616. Albouy7 noticed that the Euler- Lambert's equation mig ht have solutions not corresponding to any transfer orbits. He suggested the following formulation of Lambert's theo rem: Four functions a , r1 + r2 , r2 r1 and t are functionally dependent. Albouy gives references to different proofs of Lambert's theorem and techniques of orbit determination for t wo positions and the time of transfer: Laplace8, Adams 9, Dziobek 10, Routh11, Plummer12, Battin13. Albouy's pedantry provides a platform fo r the applied algorithm for solving Euler-Lamb ert equation. We should search for all the solutions of the Euler-Lamb ert equation. The found solutions should be checked for compliance with the sch eme of the transfer fro m the po int r1 to the point r2 for the time interval

t . It should be noted that Albouy considers the cases when the transfer from r1 to r2 takes not more than one revolution. The evaluation of the nu mber of solutions for multi-revolution transfers is given in Ref. 14 ..
Consider the algorithm for determining the transfer trajectory for t wo given positions and the time of the transfer under the condition that the transfer is performed by not more than a given number of revolutions. This algorithm is based on the methods described in the works1,2,3 and a note of Albouy7. All the solutions of Eq. (1) are searched for. Then the found solutions are che-

3


cked for co mpliance with the scheme of the transfer from point r1 to the point r2 for time interval t . The design of the algorithm uses the following facts, 1 ,2,3: ­ we al ways have: sin

2

> 0, cos

2

>0 ;
n

the sign of sin

2

is determined by the sign of cos

2

, i . e.
n n

sign sin

2

=

1, 0 1,

< <2

(3)

cos < 0 , when the second focus of the transfer ellipse is within the elliptical sector corre2 sponding to the transfer trajectory .
Let us introduce the following notation:

sin

0

2

=

r1 + r2 + r2 4a 0
0

r1

, sin ,0

0

2
0

= 2

r1 + r2 4a .

r2

r1

,
(4)

2

2

2

Figure 1 . Segment of the Secto r Does Not Include Any Foci.

Figure 2 . Segment of the Secto r Includes the Second Focus <

Figure 3 . Segment of the Secto r Includes the Second Focus >

Figure 4 . Segment of the Secto r Includes Only the First Fo cus

4


Figure 5 . Segment of the Secto r Includes Both Foci

No w consider the following possible positions of the elliptical sector corresponding to the transfer trajectory: ­ ­ ­ ­ In t there are elliptical elliptical elliptical he case A no foci within the elliptical sector, case A, Fig ure 1, sector covers only the second focus , case B, Figures 2, 3 , sector covers only the first focus, case C, Figure 4, sector covers both foci, case D, Figure 5 . = 0 , = 0 . In the case =2 = 0 or = 0,

0

. In the case C

=

0

,

=

0

or

=

0

. In the case D

=2

0

,

=

0

.

Thus, instead of Eq . (1) we can consider four equations:

t= 1

1

µ
a
3 2

a

3 2

(

0

sin

0

(
0

0

sin

0

)

+2 n

) )

(5)

t=

µ
t= 1

(

2
3 2

(

0

sin

)(
0

0

sin

0

)

+2 n

(6)

1

µ
a
3 2

a

(

0

sin

0

+

(

sin

0

)

+2 n

) )

(7)

t=

µ

(

2

(

0

sin

0

)+(

0

sin

0

)

+2 n

(8)

Among the solutions of these equations we may find some that do not correspond to transfer trajectories, i .e. those that do not start and finish in the given points. Thus we will call these four equations an extension of the Euler-Lambert equation. The solutions of the extended EulerLambert equation should be check ed for co mpliance wi th the transfer trajectories. The selection of the numerical method for solving the equations should account for the character of the functional dependence of the right parts of the equations (5)-(8) on a . Th e functions of the right parts of the equations (5) and (7), monotonically decreas e with the increase of a for n = 0 and monotonically increase for n = 1, 2, ... . The functions of the right parts of the equations (6) and (8), increase for all values of n = 0, 1, 2, ... . Thus for solving equations (5)-(8) we can use the bisection method and the golden section method.

5


The algorithm described further was used for the task of recovery of the orbital injection scheme for the case of injection with the change of incl ination14. Algorithm fo r La mbert pro blem Input information: ­ ­ ­ ­ initial position of the spacecraft; final position of the spacecraft; time interval for the transfer; maximu m nu mber of revolutions for the transfer.

rsrc
rtrg

t N max

Output information:

N

ELM

­
k

{

k

, ik ,

, ek , pk , t

,k

}



k = 1, ..., N
Wh ere
k

the number of acquired solutions (the number of reco rd s in the output array); Output array, each record contain the p arameters of the transfer orbit:

ELM

­ ­ ­ ­ ­ ­

longitude of ascending node; inclination; pericenter argu ment; eccentricity; parameter of elliptical orbit; time of passing the peri center.

ik
k

ek p
k

t

P, k

Des cription of the algorithm 1. If the vectors rsrc and rtrg are collinear, the algorithm is finished with a negative return code. Otherwise we calculate the vector m 0 =

(

0 0 mx , m0 , mz y

)

T

=

rsrc rsrc

rtrg rtrg

.

0 0 2. Calculate 2 vectors: c01 = m 0 sign mz and c02 = m 0 sign mz , 0 where sign mz

()

()

()

is the sign of the z-co mponent of vector m 0 .

The index k of the resulting array is set to zero . Items 3-5 are performed for each vector c
01

and c02 . In the description of the algorithm

these vectors will be denoted as c0 i , i = 1, 2 . Th e values calculated on their basis will have index i .as well.

6


3. Then calculate the inclination: ii = arccos c 4. The longitude of the ascending node
i
0i x

( z)
0i

0 , wh ere cz i - is the z-component of vector c

0i

is determined using conditions:

sin

i

=

c

sin ii

, cos

i

=
i

c

0i y

sin ii

(9)

5. Calculate the differen ce of true anomalies

of two given positions using the values

sin

i

and cos

i

, calculated by formulas:

cos

i

=

(

rsrc , rtrg rsrc rtrg

)

, sin

i

= c0 i , m

(

0

)

rsrc

rtrg

rsrc rtrg

.

(10)

The cycle for index i is completed. The yield is the t wo values of the difference of the true anomalies: 1 and 2. 6. Then calculat e the length s of the interval connecting the initial rsrc and final rtrg positions: . s = rsrc

rtrg .

(11)

7. Using the algorithm for solving the extended Lamb ert's equation we find two arrays of solutions:

{a {a
ond one - to
2

1, k

, ,

1, k 2, k

, ,

1, k 2, k

2, k

}, },

k = 1, ..., N

1, LEQ

k = 1, ..., N

(12)
2, LEQ

The first array corresponds to the value of the differen ce of true anomalies .

1

, and the s ec-

8. Then we calculate the unit vector corresponding to vector rsrc
0 rsrc =

(

0 xsrc ,

0 ysrc , z

0 src

)

T

=

rsrc rsrc

(13)

9. The cycle for j = 1, ..., N

1, LEQ

+N

2, LEQ

performs op erations of items 10-16 .

10. Five values are generated:

ij =

i1 , if j

N

1, LEQ j 1, LEQ

i2 , if j > N

=

1 2

, if j

N

1, LEQ 1, LEQ

, if j > N

aj =

a1,

j jN
1, LEQ

, if j
+1

N

1, LEQ

a2,

, if j > N

1, LEQ

7


j

=

1, j 2, j N
1, LEQ

, if j
+1

N

1, LEQ j

, if j > N

=

1, j 2, j N
1, LEQ

, if j
+1

N

1, LEQ

1, LEQ

, if j > N

1, LEQ

11. Then calculate the argument of latitude:

u=

0 arccos( xsrc cos

j

0 + ysrc sin j

j

),
j

if z

0 src 0 src

0 <0
and

2

0 arccos( xsrc cos

0 + ysrc sin

), if z

(14) with the values

12. Then we calculate the parameters connecting the auxiliary angles of eccentric anomaly in the initial and final points.

E2
e cos E2 p1 = ecE

E1 = E2
=1

m1

=

j

j

2
1 cos E2
, E2 p1 =
m1

(15)

rsrc + rtrg 2a
j

2 p1

(16)
m1

e sin E2 p1 = esE

2 p1

=

rsrc 2a
j

rtrg

1 sin E2

E1 + E2 2

(17)

Using the values of the sine and cosine of the h alf-sum of eccentric ano malies we can calculate the v alue of this half-sum E2 p1 . For this purpose we can use the function atan2 ( ). The values of the eccentric anomalies for the initial E1 and the final E2 points are calculated using formulas:

E1 = E2

p1

E2

m1

(18) (19)

E2 = E2 p1 + E2

m1

13. Then we calculat e: the value of the eccentricity

ej = cos

1 E1 2

rsrc + rtrg 2a E2 E + E2 cos 1 2
(20)

14. the true ano maly fo r the initial point:

= 2arctg

1+ e 1e

j j

tg

E1 2

(21)

8


15. the argument of the pericenter:
j

=u

(22)

16. time of p assing the pericenter:

t

P, j

=

a

3 j

µ

(

E1 e sin E1

)

(23)

17. parameter o f the elliptical orbit:

pj = aj 1 e

(

2 j

)
r
j ,1

(24)

18. Using the determined elements we calculate the state vectors for the times t1 = 0 and

t2 = t :

r

j ,1

and r

j ,2

. Then we calculate the residual: d j = rsrc

+ rtrg

r

j ,2

. If d j is

smaller than the set threshold value the solution is saved, otherwise - discarded. In case the solution is saved we mak e the following assignments:

k = k + 1,

k

=

j

, ik = i j ,

k

=

j

, ek = e j , pk = p j , t

P ,k

=t

P, j

19. After the completion of the cycle for j index k is used for forming the length of the output array: N ELM = k . Algorithm fo r the Extended La mbert Equation The algorithm finds the solutions of the extended Lambert's equation (1). The algorithm generates the array containing the values of the semi-majo r axis and auxiliary angles and . Th e true solutions of Lambert's equ ation, i.e. the solutions corresponding to certain transfer o rbit are among the elements of this array. The s earch for the t rue solutions requires an additional check which is p erformed by the algorithm of higher lev el. Input information:

rsrc

­ ­ ­ ­ ­

Magnitude of the vector defining the initial position of the spacecraft , Magnitude of the vector defining the final position of t he spacecraft, time interval o f the transfer, maximu m of the nu mber of revolutions for the transfer, difference of true anomalies.

rtrg t N max

Output data:

N

{

LEQ k

­

the number of elements in the array, determined values.

ak ,

,

k

,N

k

}

k = 1, ..., N

LEQ

­

We set the initial value of index k = 0 .

9


The operations 1-3 are performed in the cycle for N from the minimum possible value of the semi-major axis a = and the angle 0 <
0

0to N max 1 . We calculate
of the transfer trajectory

(

)

rsrc + rtrg + s 4

<

, determined by the equality:

sin

0

2

=

rsrc + rtrg 4a

s

(25)

The minimu m value o f the semi-major axis can be attained only for the case boundary elliptical sector. In this case the segment connecting the ends of radius- vectors r1 and r2 , includes the second fo cus of the ellipse, the angle = . If , we check the equ ality:

µ t=a

3 2

(

0

sin

0

)

+2 N
=,=
k 0

(26)

If the equality is satisfied we save in the output array the values: a, ing this we add 1 to the index k and make the assignments: ak = a, After that we return to the beginning of the cycle with new value o f N . If

, N . For do0

=,

k

=

, Nk = N .

<

2 , we check the equality:

µ t=a

3 2

+

(

0

sin

0

)

+2 N
k

(27)

If the equality is satisfied we make the assignments: k = k + 1 , ak = a ,

=

,

k

=

0

,

N = N k and we go to the beginning of the cycle with new v alue of N .
If none of the above mentioned equalities are satisfied we go to the next item 2 . 1.

+ a to amax we search for the zeroes of 4 two functions whose shape depends on the value of th e angle . Here the parameter a
is of the ord er of unit meters, and amax does not exceed 300 ,000 km. If , we search fo r the zero es of functions:
1

Within the interval fro m amin =

rsrc + rtrg + s

( a)

= µt a
3 2

3 2

0

sin

0

(
0

0

sin 0 + 2 N ,
0

)

(28)

2

( a)

= µt a

2

(

0

sin

)(

sin

0

)

+2 N .

If

<

2 , we search for the zeroes of functions:

10


3

()

a= µt a
3 2

3 2

0

sin

0

+

(

0

sin

0

)

+2 N ,
(29)
0

4

( a)
sin

= µt a

2

(

0

sin

0

)+(
=

0

sin

)

+2 N .

Wh ere
0

2

=

rsrc + rtrg + s 4a

, sin

0

rsrc + rtrg 4a

s

2

. we find the value a

(30) ,

Thus, using the bisection algorithm within the interval amin , amax for which the function
k

=0

is equal to zero .
0

3. Then we calculate the angles 0 <

< , 0<
, sin
0

0

<

, determined by the equ alities:

sin

0

2

=

rsrc + rtrg + s 4a
=0

2

=

rsrc + rtrg 4a
=0

s

. and :

(31)

Dep ending on the nu mber of the function k we calculate the angles

k
1 2 3 4 ing values of
k

=0

=
0

0

2 =
2
0

= = =

0 0 0 =0

0

In the output array we sav e the found value of the semi-major axis a

and the correspond=0

, , N . For this we make the assignments : k = k + 1, ak = a

,

k

=,

= , Nk = N .

4. After completion of the cycle the index k is used for forming the length of the output array: N LEQ = k . Evaluation o f the energy costs for co planar tra nsfers. Geo metrical metho d For maneuver characterization problems we al ways k now the state v ectors for the spacecraft before and after the maneuver. In the scope o f spacecraft control theory these p roblems are called rendezvous problems. Thus we should use the techniques for solving rendezvous problem as the basis for the maneuver characterization problem. In the case of the non- p erturbed motion this is the Lambert problem. Ho wever, for ev aluation of the results we should compare the obtained energy costs with the case for which we have the parameters of the targ et orbit, but the position in this orbit is not defined. Wh en we control the spacecraft for reaching certain position for a given time we perfo rm phasing maneuvers which provides such conditions for the orbital transfer that the energy cost for the orbital transfer with the set position in the target orbit does not signifi-

11


cantly differ from the orbital transfer with an arbitrary position of the spacecraft in the target orbit after the transfer man euver. Let us consider the geo metrical method of the transfer fro m one coplanar o rbit to another. Consider the family of ellipses with the focus in the origin of coordinates. This family of ellipses is described by the parameters l and f which are determin ed by the following relationships

c=

f l c , a= , e= = , 2 aa 2

f

(32)

where c ­ half distance bet ween the foci a ­ semi-major axis e ­ eccentricity If the p ericenter is placed to the left of the coordinate o rigin then f > 0 , otherwise f < 0 The distance to the spacecraft and the velocity at apocenter r , V have the following relationships with l and f :

.

and pericenter r , V

r= r=

l+ f 2 l 2 f

,V= ,V=

2µ l f , l l+ f 2µ l + f , llf

(33)

where µ is the gravitational constant. Each orbit is represented by a point in the semi-plane l , f , l > 0 . If f > 0 , we have apocenter right from the origin of coordinates, if f < 0 - pericenter. We will consider applying the burn pulses only at pericenter and apocenter. Let us consider the orbital transfer from the point l1 , f1 to the point l2 , f 2 plied at the apsidal point right-side from the origin of coordinates, the orbital the distance to this apsidal point. For f1 > 0, f 2 > 0 - this is an orbital tran apocenter will remain right-side from the origin of coordinates before and after . If tran sfer the the burn is apsfer must keep for which the burn , thus (34)

l1 + f 2

1

=

l2 + f 2

2

The case f1 > 0, f 2 < 0 correspond to the orbital transfer for which we will have the pericenter right-side fro m the origin of coordinates after the bu rn is applied. Thus:

l1 + f1 2

=

l2 2

f

2

=

l2 + f 2

2

(35)

Consideration of the cases f1 0, f 2 < 0 and f1 0, f 2 > 0 , yields that the application of the burn on the right-side of the o rigin of coordinates keeps the value l + f .

12


Similar considerations on the cases when the bu rn is applied left-side from the o rigin of coordinates will result in the conclusion that for these cases the value l + f is kept. Thus the broken line in the semi-plan e l , f , l > 0 , corresponding to the sequence of orbital transfers will consist of segments with inclination angle tangents -1 and +1 (Fig. 6). The value -1 for the inclination angle tangent corresponds to the burn applied right-side from the origin of coordinates and the value +1 ­ corresponds to the left-side application of b urn. The v elocities in the apsidal points (in the apocenter or pericenter) right-side Vr and left-side Vl from the origin of coordinates are calculated using formulas:

Vr = Vr = 2 µ ( l2 f2 )

2µ(l f ) , Vl = l (l + f ) 2 µ ( l1 f1 ) , Vl =

2µ(l + f ) . l (l f ) 2 µ ( l2 + f 2 ) l2 ( l2 f2 ) 2 µ ( l1 + f1 ) l2 ( l1 f1 ) .

(36)

l2 ( l2 + f 2 )

l1 ( l1 + f1 )

(37)

For the orbital transfer from the point l1 , f1 to the point l2 , f 2 the values of the transfer impulses Vr (when the burn is applied right-side fro m the origin of coordinates) and Vl (when the burn is applied left-side fro m the origin of coordinates) are calculated using the formulas:

Figure 6 . Orbital Transfers D epict ed In the SemiPlane l , f , l > 0 . Fo r the Transf er From the Point l1 , f1 To the Po int l2 , f 2 The Burn is Applied R ight-Side From the Origin Of Coordinates, For the Transfer Fro m l2 , f 2 To the Point

Figure 7 . Hohmann's Transf er Plotted In the Semi-Plane l , f , l > 0 . The Upper Broken Line Corresponds To the First Burn Applied LeftSide From the Orig in of Coordinat es, the Low er Line ­ To the Right-Side Application.

l3 , f3 ­ To the Left Side
Let us consider the Hohmann's transfer fro m the circular orbit with radius l1 to the circular orbit with radius l2 (Figure 7). The elliptical transfer orbit will have the p arameters:

l3 =

l1 + l2 2

, f3 = ±

l2 2

l1

. The sign « +» correspond to the burn applied to the left-side of the

13


origin of coordinates and the sign «-» - to the right-side application. Using (37) for l2 will obtain the consumption of the characteristic velocit y for the Hohmann's transfer:

> l1 , we

VH =

2µ C l1

()

,

C

()

=

2 1+

1+

1

(

2

+1

)

,

(38)

where

=

l2 l1

.

The graph o f the function C

()

is shown in Figure 8.

Let us consider the three burn bi-elliptical transfer from the circular orbit with radius l1 to the circular o rbit with radius l2 (Figure 9). The first burn is applied left-side fro m the origin of coordinates and makes the semi-major axis of the first transfer orbit where times greater, i.e. l

p1

= l, 1

> 1 . Parameters of intermediate orbits l p1 , f p1 , l p 2 , f

p2

are connected by the followin g

relationships:

l1 2

=

l

p1

f 2

p1

,

l

p1

+f 2

p1

=

l

p2

+f 2

p2

,

l

p2

f 2

p2

=

l2 2

.

(39)

Figure 8 . Graph Of the Funct ion C

()

.

Figure 9 . The Three Burn Bi- Elliptical Transfer From the Circula r Orbit With Radius l1 To the Circula r Orbit With Radius l2 and the Hohmann's Transf er. Hohmann' s Transf er Corresponds To the Broken Line Including the Points: (l1,0) - > ( l3,f3) -> (l2,0). The Three Burn Bi- Elliptical Transfer Co rresponds To the Broken Line Including the Po ints: (l1,0) -> (lp1,fp1) - > (lp2,fp2) -> ( l2,0).

Solving the system of equations (39) with respect