orbital mechanics: 7. relative motion in orbit

7 Relative Motion in Orbit

7.1 Space Rendezvous Rendezvous in space between two satellites is accomplished when both satellites

attain the same position and velocity, both vectors, at the same time. However, at the time a rendezvous sequence is initiated, the two satellites may be far apart in significantly different orbits. In fact, one satellite may be starting with a launch from the ground.

This chapter will address the rendezvous sequence in two parts. The first part will be concerned with phasing for rendezvous, i.e., developing the maneuvers and timing sequence that will bring the two satellites into close proximity. The material presented in the sections dealing with Hohmann and bi-elliptic transfer is based on the approach presented in Ref. 1. The second part, terminal rendezvous, will examine the motion of one satellite with respect to the other in a coordinate frame attached to one of the satellites. Relative motion between the satellites and terminal maneuvers required for docking will be examined.

Phasing for Rendezvous H o h m a n n t rans f e r . The requirements for rendezvous between two satellites

in circular coplanar orbits are both illustrative and operationally useful. Figure 7.1 presents a sketch of two circular orbits with radii r i and r f . Assume the satellite in the inner orbit to be the active rendezvous satellite, i.e., the maneuvering satellite. The satellite in the outer orbit is the passive target satellite, i.e., nonmaneuvering. Further, assume that, at some instant in time, the rendezvous satellite is located at the point shown in Fig. 7.1 and the target satellite is located ahead, i.e., in the direction of motion, in its orbit by an amount equal to the central angle On.

Now, assume that the rendezvous satellite initiates a Hohmann transfer in order to rendezvous with the target satellite at the rendezvous point. If travel times for the two satellites are equated,

Ptr 7C--OH - - zr - - 2 - P f (7.1) 2

where

Ptr = the orbital period of the Hohmann-transfer ellipse P f = the period of the target satellite orbit

Substituting for both periods,

(7.2)

135

136 ORBITAL MECHANICS

~Vf Rendezvous point

A O

r ~Vl Initial f target location

s a t e l l i t e

Fig. 7.1 Rendezvous via Hohmann transfer.

Reducing and solving for O H ,

(7.3)

The range of OH is

0 ~ 0 g ~ 2"/" 1 - - = 0.64645Jr = l16.36deg

Figure 7.2 presents OH as a function of the final orbit altitude hf for an initial orbit altitude hi of 100 n.mi. (185.2 km).

If the initial lead angle of the target satellite with respect to the rendezvous satellite is not OH but, instead, is OH + A0, then the Hohmann transfer cannot be initiated immediately. If it were initiated immediately, then the target satellite would be located an angle A0 beyond the rendezvous point when the rendezvous satellite reached the rendezvous point. And so the initiation of the Hohmann transfer must wait until the phase angle reduces to OH. This will occur naturally because the angular velocity of the inner orbit wi is higher than the angular velocity of the outer orbit coy.

In time t, angular displacements of the two satellites will be Oi = coit and O f = o)ft. Therefore, AO = Oi - Of = (co i - cof )tw, where tw is the waiting time to achieve a phasing angle change A0. The maximum value of tw is the synodic period Ps when A0 = 2zr

2Jr 2zr Ps - - - -- (7.4)

coi -- w f (2rc/Pi) - ( 2 ~ / P f )

RELATIVE MOTION IN ORBIT 137

60

50

40

3O

~= 20

10

Fig. 7.2

/ / S

/ J

I000 2000 3000 4000 5000 6000 hf, km

Phase angle for rendezvous via a Hohmann transfer from a 185-km orbit.

o r

1 1 1 - - (7.5)

Ps Pi P f

Figure 7.3 presents P~ vs hf for hi = 100 n.mi. (185.2 km). Note that, if hf = 120 n.mi. (222.2 km), the synodic period is approximately 10,000 min, or about one week. The possibility of such long waiting times will be circum- vented in the next section by using hi-elliptic and semitangential transfers instead of a Hohmann transfer.

For the Hohmann-transfer technique, the total time for rendezvous, t, is the sum of the Hohmann-transfer time t H and the waiting time tw,

A0 A0 t = tn + tw = tn + - - -- tn + Ps (7.6)

( .0 i - - ( . O f

When the second AV of the Hohmann transfer is applied by the rendezvous satellite, both satellites will have the same velocity at the rendezvous point at the same time, and rendezvous will be accomplished.

Bi-elliptic transfer. In Chapter 5, it was concluded that, in terms of A V, the bi-elliptic transfer is not significantly better than the Hohmann transfer. However, for rendezvous, the bi-elliptic transfer will be shown to have utility in the case for which the Hohmann transfer is weakest, i.e., for waiting times approaching the synodic period.


z

10,000

9000 8000

6000

5000

4000

2000

900 800

600 N

300

\

100

200

Fig. 7.3

300 500 IO00 2000 3000 5000 h f , ktn

Synodic period for a 185-km inner orbit.

Figure 7.4 presents a sketch of the bi-elliptic transfer previously discussed in Chapter 5. In this case, rendezvous will occur at the rendezvous point after the application of A V3.

Assume the target satellite initially at an angle (OH + AO) ahead of the rendezvous satellite. Since the radius rt is assumed to be greater than r f , the target satellite must first traverse 27v - A0 -- OH, and then 2re, in order to reach the rendezvous point at the same time as the rendezvous satellite. This total time is

But

from Eq. (7.1), so that

27r - AO 2:r - OH t -- - - + - - (7.7)

O)f (.Of

P~ ~ - OH

O)f 2 ' 27r P f = tH

t - - m

2~r - AO P f 2yr PU + tu + -~-

RELATIVE MOTION IN ORBIT

& v z

139

r' I

rf /hVa

Initial target location

Fig. 7.4

& v 3 Rendezvous point

Rendezvous via the bi-elliptic transfer.

and

At A0 = 0,

And at A0 = 2Jr,

(7.8)

3 t = t . + ~e l

t = tt4 + PU 2

Thus, because the bi-elliptic transfer occurs mostly beyond the outer circular orbit, it easily accommodates a A0 that is slightly less than 2zr. In this case, rt will be only slightly larger than r f .

In all cases, the value of rt is determined by the value of A0 because the time spent by the rendezvous satellite in the elliptic-orbit transfer legs is

- J i l l 3" t - - - - ~ + (7.9)


T ime

5Pf t, + --g-

tN+ ts

tL

t . + -~

t .

-~--HOHMANN TRANSFER

0 AOi 2'n-

A8

Fig. 7.5 Total time vs phase angle for the Hohmann and bi-elliptic transfers.

Figure 7.5 presents total time vs A0. The Hohmann line is specified by Eq. (7.6). The bi-elliptic line is specified by Eq. (7.8). Note the difference in slope. The total time for the bi-elliptic transfer is a minimum at A0 = 2rr. For smaller values of A0, the total bi-elliptic time becomes longer. Therefore, there is no advantage to be gained by waiting because waiting reduces A0, which increases the total bi-elliptic time.

Figure 7.5 depicts an intersection of the Hohmann and bi-elliptic lines. An intersection will exist only if Ps > Pf/2. Substituting for Ps,

> - - (7.10) P~.-P~- 2

or

3Pi > PU (7.11)

Since period P = 2rrr3/2/~r~, then,

rf < 32/3ri (7.12)


Since 32/3 is approximately 2.08, Eq. (7.12) determines that, for an initial orbit altitude of 100 n.mi. (185.2 km), the limit of usefulness for the bi-elliptic phasing technique is a final orbit altitude of 3927.7 n.mi. (7274.1 km). For final altitudes above this value, the Hohmann-transfer technique should be employed for all values of A0.

When there is an intersection at AOi, the Hohmann technique would be used when 0 < A0 < AOi, and the bi-elliptic technique would be used when AOi <_ A0 _< 2rr. To find AOi, set the total Hohmann time to the total bi-elliptic time

tH +

Solving for AOi,

AOi

(2rr/Pi) - (27r/Pf) (7.13)

The corresponding total time is

(7.14)

3/, ti = t14 + - - (7.15)

2

The break-even phasing angle AOi is presented as a function of h f for hi = 100 n.mi. (185.2 km) in Fig. 7.6.

Somitangontial transfer. One more transfer technique to achieve coplanar rendezvous should be examined. Figure 7.7 illustrates the semitangential transfer. The rendezvous satellite achieves a transfer ellipse by applying a A V1 that is larger than A V1 for a Hohmann transfer. This transfer ellipse intersects the final circular orbit at two points, 11 and 12. Rendezvous can be accomplished at either point by the application of a second A V to circularize the orbit.

The rendezvous solution proceeds as follows:

1) Apply a specified 2xVa to Vci in the direction of motion. 2) Knowing the perigee radius rp and the perigee velocity Vp of the transfer

ellipse, calculate the semimajor axis a, for the transfer ellipse from the energy equation

3) Calculate the eccentricity e of the transfer ellipse from

rp = a(1 - e) (7.17)

4) Calculate the true anomaly vl of the intersection point, 11, from the orbit equation

a(1 - e 2) (7.18)

rf -- 1 + e c o s v l


360

300

240

180

120

60

0

0

/ /

/ /

J J

rf = 32/3 r i

hf = 7274.1 km

rf = 13652. 3km

J

i000 2000 3000 4000 5000 6000 7000 h i , km

Fig. 7.6 Break-even point phasing angle for hi = 185 km.

For 12, v2 = 360 deg - v l , 5) Calculate the flight-path angle y~ from

e sin vl tan )/1 -- (7.19)

1 + e cos vl

For 12, )/2 = - Y l , 6) Calculate the eccentric anomaly E1 from

e + cos v 1 cos E1 -- (7.20)

1 + e cos u 1

7) Calculate the period Pt of the transfer orbit from

27ra3/2 P -- (7.21)

8) Calculate the t ime tl f rom the applicat ion of A Vl to the intersection point I1 from Kepler ' s equation,

P tl = ~ (El - e sin E l ) (7.22)

8000


Transfer ellipse

AV 2

12 [ ql rf

, '1 r. i

i V1

I 1

Initial location.of the J rendezvous satellite

Fig. 7.7 Semitangential transfer geometry.

Vc± AV I

For 12, t2 = P - tl, 9) Calculate A01 for the first intersection point 11 from

(360 deg)q 0/4] mod(27r) (7.23) zX01 = vl PI

For I2, calculate A02 from Eq. (7.23) using v2 and t2. 10) Calculate A V2 from the vector triangle described in Fig. 7.7

AV2= ffV 2 + V~- 2V1VcfcosF1 (7.24)

The magnitude of A V2 is the same for I1 and 12 because the vector triangles are congruent.

The previous semitangential transfer was tangent to the initial orbit at the point of departure. A different semitangential transfer is tangent to the final orbit at the point of arrival. The solution for this technique proceeds by selecting a value for the semimajor axis of the transfer orbit. Because the apogee radius equals the radius of the final orbit, the transfer-orbit eccentricity can be calculated from Eq. (7.17). Similarly, Eqs. (7.18) through (7.22) can be used to calculate conditions at the departure point. Then,

(360 deg) ( ~ Z - - t l ) A01 = (180 deg - Vl) - - 0/4 (7.25) Ps


Finally, A V1 is calculated from

AV1 =~/V? q- Vc2 - 2V1VcicOSyl (7.26)

where the subscript i refers to the point of departure. AV2 is simply the difference between the circular orbit velocity of the final orbit and the apogee velocity of the transfer orbit.

Authors' note. The authors are grateful to A. S. Ganeshan and R. Gupta, of the ISRO Satellite Centre in Bangalore, India, who pointed out a deficiency in the semitangential transfer results presented in the first edition of this book.

Ganeshan and Gupta also pointed out the usefulness of solutions to Lambert's problem. These solutions should, of course, be included and compared. This second edition includes results for the Lambert, Hohmann, bi-elliptic, and semitangential with tangency at departure techniques. Also included are results described earlier in this section for the semitangential with tangency at arrival technique. Comparisons among these techniques are quite interesting.

Transfers based on solutions to Lambert's problem. Lambert's problem, namely, to find the transfer orbit that connects two given position vectors in a specified transfer time, has been the subject of extensive literature. Pitkin, 2 Lancaster and Blanchard, 3 Herrick, 4 Battin, 5 Gooding, 6 and Prussing and Conway, 7 among others, have made important contributions to the solution of the problem.

Lambert's theorem states that the transfer time depends only on the semimajor axis of the transfer orbit a, the sum of the radii to points 1 and 2, (rl + r2), and the chord length c between the points. Figure 7.8 presents the geometry of the transfer orbit. Note that the transfer-orbit eccentricity is not explicitly included.

2a

2

J Fig. 7.8 Transfer-orbit geometry.

l


Lambert's equation, the solution to the problem, is derived very clearly by Prussing and Conway, 7

where

V/-ff(t2 - q) = a3/2[ot - fl - (since - sin fl)] (7.27)

r l + r 2 + c = 2 a ( 1 - - c o s o t ) = 4 a s i n 2 ( 2 ) (7.28)

r l + r 2 - c = 2 a ( 1 - - c o s f l ) = 4 a s i n 2 ( ~ ) (7.29)

Ambiguities, indeterminancies, and the transcendental nature of Lambert's equation complicate the process of solving for actual values. Herrick 4 developed a universal solution that co-author C. C. Chap developed into a computationally robust and efficient software routine.

Solutions to Lambert's equation are very important to phasing for rendezvous. By letting rl and r2 correspond to the initial- and final-orbit radii and by relating c and A V geometrically through

c = v/r 2 + r 2 - 2rlr2 cos Av (7.30)

the following procedure was used to calculate transfer solutions by Lambert's equation.

For this technique, the equation for A0 is

(360 deg) At A0 = Av 0~/ (7.31)

Pf where Av = v2 -- Vl = transfer arc and At = t2 - tl = transfer time. Then, use the following procedure:

1) Select a value for A0. 2) Select a value for Av, and calculate At from Eq. (7.31). 3) Use the HERRIK or similar solution routine to calculate transfer-orbit char-

acteristics, including the transfer-orbit velocities at points 1 and 2. Because rl and r2 correspond to initial and final circular orbit radii, the routine will also calculate AV1 and A V 2 required for the transfer. Finally, AVT = AV1 -~- AV2.

By cycling through many values of Av for a specified A0, a minimum value of AVr can be found iteratively. Then, by repeating this process for many values of A0, a curve of minimum A VT Lambert solutions can be developed. These solutions should be very efficient of A Vr because they are unconstrained with respect to the direction of A V application.

Comparison of transfer techniques for coplanar rendezvous. Compar- isons among rendezvous techniques usually take the form of comparisons of required A Vr and elapsed time as a function of initial geometry. A comparison of the Hohmann-, bi-elliptic-, semitangential-, and Lambert-transfer techniques for


14,000

12,000 h i - 185 km hf = 556 km

o~ 10,000 !

d E

8000 0.

0

~oo

Lambed

[ ~ Tangent at departure using 12 .. 4000 Lambed and

- Tangent at arrival Tangent at departure using 11

2000 Tangent at arrival

00 40 80 120 160 200 240 280 320 A0, ~ Degrees

360

Fig. 7.9 Total elapsed time for rendezvous as a function of initial position of the target satellite for six rendezvous phasing techniques.

coplanar rendezvous will now be made for two specified circular orbits, hi = 100 n.mi. (185.2 km) and hf = 300 n.mi. (555.6 km).

Figure 7.9 presents an actual version of Fig. 7.5 for six transfer techniques. Total elapsed time for rendezvous is presented as a function of A0. The Hohmann and bi-elliptic curves cross at AOi = 42.7 deg, as can be verified by Fig. 7.6. The associated time ti from Eq. (7.15) is 10,700 s. Only a portion of the Hohmann- transfer curve is shown as it extends off-scale. This curve continues linearly to a value of 69,700 s at A0 = 360 deg. This value is simply the Hohmann-transfer time (2759 s) plus the synodic period (66,941 s) from Eq. (7.6) and Fig. 7.3. The bi-elliptic curve varies linearly from 11,378 s at A0 = 0 to 5632 s at A0 = 360 deg.

For the large middle region of A0 between approximately A0 ~ 10 deg and A0 = 340 deg, the semitangential and Lambert solutions are represented by a pair of curves whose values are very nearly the same. At A0 = 0, the time for the Lambert transfer is 10,801 s, and the time for the semitangential transfer is 10,807 s. At A0 = 340 deg, the Lambert-transfer time is 4536 s, whereas the semitangential- transfer time is 4633 s. In this large A0 region, the best semitangential transfers are tangent at departure.

Figure 7.10 presents the total transfer A V requirement as a function of A0 for the six transfer techniques. The Hohmann-transfer curve for A Vr is a horizontal line at a value of 211 rn/s. The bi-elliptic curve for AVT ranges from 2782 m/s at A0 = 0 to 211 m/s at A0 = 360 deg.

The curves for the Lambert and semitangential transfers are very close. The difference in values for A Vr between the two techniques is 5 rn/s or less for


3500

Lambert and hi = 185 km 3000 ~ ~ d e p a r t u r e using 12 h i = 556 km

2500 ~ ~ " ' ~ , ~ Tangent at departure using I

000-, F- Bi-elliptic

> 1500 --

_ 1000

500 _ ~ L a m b e r t and Tangent at arrival Hohmann

I I m m I I m I 00 40 80 120 160 200 240 280 320 360

AO, ~ Degrees

Fig. 7.10 Total A Vr required for rendezvous as a function of initial position of the target satellite for six rendezvous phasing techniques.

10 dog _< A0 _< 340 deg, with the Lambert values being the better, that is, the smaller one.

Comparing all six techniques on both Figs. 7.9 and 7.10 in the range 10 dog < A0 _< 340 dog, the Hohmann technique is best in terms of time and A Vr for small values of A0. As A0 increases, the Hohmann time increases rapidly while the times for the other techniques decrease gradually. As A0 increases, the A Vr values for the bi-elliptic, semitangential, and Lambert techniques gradually decrease. In this large middle region of A0, comparative values of time and AVr must be evaluated for each specific application. If A Vr is critical and time is available, then the Hohmann technique is attractive. If time is critical and A Vr is available, then the Lambert and semitangential techniques are attractive.

Results in two regions, 0 < A0 < 10 dog and 340 dog _< A0 _< 360 deg, are very interesting and need to be examined closely. Figures 7.11 and 7.12 present time and A Vr vs A0 for 0 < A0 ~ 9.5 dog. Curves are presented for the Hohmann, Lambert, and semitangential techniques. This semitangential technique is different from the previously discussed semitangential technique in that this transfer is tangent at arrival. This Lambert solution is very similar to the semitangential transfer and is different from the other Lambert solutions in this A® range shown on Fig. 7.9 and 7.10 in that the times and AV are much lower. All of the solutions have common values at A0 = 0, namely, At = 2759 s and AVr = 211 m/s. As A0 increases, the Lambert and semitangential curves, which are virtually indistinguishable, display shorter travel times and higher AVr values than the Hohmann solutions. The Lambert solutions are slightly shorter in time, 35 s or less, and slightly lower in AVr, 0.5 m/s or less, than the semitangential solutions. In both solutions, the active satellite establishes a transfer trajectory whose perigee altitude is less than 185 km and whose apogee altitude is at or very near 556 krn. The transfer arc is slightly more than one-half revolution. An unfortunate


5 0 0 0 I I I I I I I I 1 I ~

J h i = 185 km 4 6 0 0 - - hf = 556 km

o H0hmann

(n 4 2 0 0 - -

-o 3 8 0 0 - - Tangent at arrival* - - G) ¢) ¢1.

N 8400 Lambert

3 0 0 0

26oo I I I I I I I I I I I 0 1 2 3 4 5 6 7 8 9 10 11 12

AB, ~ Degrees

Fig. 7.11 Total elapsed time vs A 0 for 0 _< A 0 < 9.5 deg for three rendezvous phasing techniques.

characteristic of these transfer trajectories is that the perigee altitude decreases as A0 increases. For A0 = 6 deg, the Lambert solution perigee altitude is 120 km and, for A0 = 9.5 deg, the Lambert perigee altitude is 55 km. Therefore, Lambert and semitangential solutions beyond about A0 = 6 deg are impractical because the active satellite would have to negotiate a very low perigee altitude. For 0 < A0 < 6 deg, the choice between the Lambert and Hohmann techniques is the classic tradeoff between shorter transfer times and higher velocity requirements.

Transfer solutions in the range 340 deg < A0 _< 360.1 deg are displayed on Figs. 7.13 and 7.14. Figure 7.13 presents travel time for the Lambert solution and both semitangential solutions, that is, tangent at departure and tangent at arrival. In addition, tangent at departure solutions are presented for both intersection points, 11 and 12. See Fig. 7.7. Travel-time curves for the Hohmann and bi-elliptic solutions are beyond the scale of this figure.

The curves on Fig. 7.13 display some unusual characteristics. As A0 decreases from 360 deg, values for At increase for tangent at departure using intersection 12 and decrease for tangent at arrival and for tangent at departure using 11. As A0 decreases, values for At decrease and then increase for the Lambert technique. Figure 7.14 shows equally surprising results for A Vr vs A0 for these techniques.

In the range 355 deg < A0 _< 360 deg, the tangent at arrival solutions provide fast transfers for reasonable values of A Vr. At A0 = 356 deg, the tangent at arrival solution is At = 1940 s and A Vr = 366 m/s. The central angle of travel in the transfer orbit is only 122 deg from departure to arrival at apogee. The perigee altitude is only 90 km and is less as A0 decreases, but the satellite does not traverse the perigee region in these solutions. The detracting feature of these solutions is that A Vr increases rapidly for A0 < 355 deg. However, these solutions are attractive for emergency rendezvous missions where transfer time is critical.


450 I I I I I I I I I I

400 -- h i = 185 km Tangent at a r r i v a l * ~ -- hf = 556 km

"-~ 3 5 0 - ~ Lambert* _

F- > 3 0 0 - <

250

200 I I I I I I H°hr~ ann I I 0 1 2 3 4 5 6 7 8 9 10 11

Ae, ~ Degrees

Fig. 7.12 Total A V r vs AO for 0 < AO < 9.5 deg for three rendezvous phasing techniques.

5000 ' t ' t ' t ' l ' I ~ t ' I

5000 ~ ~ , ~ T a n g e n t at departure using 12

3600 - O "i = 185 km \ \ ¢/)

hf = 556 km hf = 556 km ~ 3200

2800

2400 •

Tangent at departure using I1 #s ,~ , . . ~ i

2000 e/ ~ - d Tangent at arrival/•

• i 16oo , I t I , I , I , I , I t I

346 348 350 352 354 356 358 360 Ae, ~ Degrees

Fig. 7.13 Total elapsed time vs A 0 for 346 < A 0 < 360.1 dog for four rendezvous phasing techniques.


E

>

Fig. 7.14

800 , I ' I ' I ' I ' I ' I ' I ' I ' I J I

~ , ~ h i = 185 km Tangent at - 700 ~ , ~ . ~ hf = 556 km departure using 11 ~ ' - -

600 . , ~ , , , , ~ , , ~ T a n g e n t at d e p a r t u r e using 12 • - -

400-- ~ ~ ~ i Bi-elliptic ~ , ~ ~

- ./Hohmann ~ l

200 I I I i I j I I , I i I J I i I j ",'1 340 342 344 346 348 350 352 354 356 358 360

/~O, - Degrees

Total AVT vs 340 < ~ 0 < 360.1 dog for six rendezvous phasing techniques.

Tangent at departure solutions using 11 are available for emergencies in the narrow range 359 deg < A0 < 360.1 deg. As the figures show, these transfers are very fast, that is, of short duration. However, the AVr rises very quickly as A0 increases to 360.1 deg and then decreases to 359 deg.

In terms of A Vr, the Lambert solutions are always lower, although only slightly so, than the tangent at arrival solutions. In terms of At, the Lambert curve decreases as A0 decreases, but then it levels out at A0 ~ 357 deg and increases thereafter. For A0 < 355 deg, the Lambert solutions more closely resemble the tangent at departure solutions using/2 than the tangent at arrival solutions. The Lambert solutions are always shorter in time and lower in A Vr than the tangent at departure solutions using I2.

Finally, it is interesting to note that the Lambert and tangent at arrival solutions are lower in A Vr than the bi-elliptic solution for 358.5 d e g < A0 < 360 deg. And no solutions are lower in AVr than the Hohmann solution of 211 rigs. All of the solutions coalesce to exactly the same solution at A0 = 360 deg.

Three-Dimensional Space Rendezvous Modi f ied Hohmann-t ransfer technique. Figure 7.15 depicts the modified

Hohmann-transfer technique for three-dimensional rendezvous. Three-dimensional rendezvous means that the initial and final circular orbits are not coplanar but have a dihedral angle of rotation between the orbit planes a, as shown on the figure. Thus, a Hohmann transfer and a plane change maneuver are required for this technique.

Phasing is accomplished in the same way that was described for the coplanar Hohmann-transfer technique, except that the phasing angles are measured in two different orbit planes. Figure 7.15 shows the line of intersection of the two planes as the line of nodes. Let the in-orbit positions of the satellites be measured from this line, i.e., Oi describes the position of the rendezvous satellite in the initial orbit,

RELATIVE MOTION IN ORBIT

FINAl OR BIT-~.

151

.HOHMANN TRANSFER

NITIAL ORBIT

NDEZVOUS POINT

/--LINE OF NODES

Fig. 7.15 Modified Hohmann-transfer maneuver.

and Of describes the position of the target satellite in the final orbit. Let 0i - Of = On + A0, as in the coplanar case. After a waiting time, A0 becomes zero, and the Hohmann transfer is initiated. When the rendezvous satellite circularizes into the final orbit, both satellites are equidistant from the rendezvous point. When they simultaneously reach the rendezvous point, the rendezvous satellite performs a single-impulse plane change maneuver to rotate its orbit plane through the angle o~, and rendezvous is accomplished. The time required for this technique is the sum of 1) the waiting time to achieve A0 = 0, 2) the Hohmann-transfer time, and 3) the time required for the rendezvous satellite to traverse the final orbit from the circularization point to the line of nodes.

Figure 7.16 describes the total velocity A Vr required for the three impulses as a function of the plane change angle ot and the final circular orbit altitude hf when the initial circular orbit altitude hi = 100 n.mi. (185.2 km). When ot = 0 and hf = 300 n.mi. (555.6 km), AVT = 211 m/s. This corresponds to the value shown on Fig. 7.10 for the coplanar Hohmann transfer. Note that AVr increases very rapidly as ot increases.

Bi-elliptic transfer with split plane changes. This transfer technique was previously described in Chapter 5. For three-dimensional rendezvous, the bi- elliptic transfer is initiated when the rendezvous satellite reaches the line of nodes. As in the coplanar case, the value of A0 determines the altitude ht of the intermediate transfer point. For the example of hi = 100 n.mi. (185.2 km) and hf = 300


3000

20 2soo \ \ \ -" "- / -

1000 ' • ,-,

~ ~ 1 ~ / /1500 h km 5 O0 ~ ~ 0 £ 0 f '

o oo

>5000

Fig. 7.16 Velocity required for a modified Hohmann transfer from a 185-km parking orbit.

n.mi. (555.6 km), Eqs. (7.8) and (7.9) determine h, as a function of A0. Then, given values for hi, ht, hf , and the total plane change angle otr, Ref. 8 describes the optimal plane change split, i.e., o~1, ~2, o~3, to minimize the total AV. Figure 7.17 presents these solutions for AVT as a function ofo~r and A0 for hi = 100 n.mi. (185.2 km) and h / = 300 n.mi. (555.6 km). Because the AVr for the coplanar hi-elliptic transfer is large compared to the Hohmann transfer (see Fig. 7.10), the advantage of the optimal split-plane change for the three-dimensional bi-elliptic transfer produces smaller values of A V:~ than the modified Hohmann transfer only for large values of A0.

In-Orbit Repositioning Maneuvering technique. I f a satellite is to be repositioned in its circular

orbit, this maneuver can be performed by applying an impulsive velocity along the

4500

4000

3500

3000

2500

2 0 0 0

1500

i000

500

0

Fig. 7.17

a T - DE 1~, / ~

6 B, DEG

zT-- / / Z

]30~

/

d

Velocity increment AVT necessary for a bi-elliptic transfer from a 185-km to a 556-km circular orbit.


velocity vector, either forward or retro. With a forward A V1, the satellite will enter a larger phasing orbit. When the satellite returns to the point of A V application, it will be behind its original location in the circular orbit. The satellite can re-enter the circular orbit at this point by applying a retro A V equal in magnitude to the first A V. In a sense, a rendezvous with this point has been performed. Or the satellite can remain in the phasing orbit and re-enter the circular orbit on a future revolution. The satellite will drift farther behind with each additional revolution.

If the first A V is in the retro direction, the satellite will enter a smaller phasing orbit and will drift ahead of its original location in the circular orbit. The drift rate, either ahead or behind, is proportional to the magnitude of the A V.

Application to geosynchronous circular orbit. A very common application of repositioning is the drifting of a satellite in a geosynchronous circular equatorial orbit from one longitude to another. The change in longitude is given by the equation

A L = LnPpH

where

AL = the change in longitude L = the drift rate, positive eastward n = the number of revolutions spent in the phasing orbit PPH = the period of the phasing orbit

The drift rate L is given by

(7.32)

( Pr,~ ~ Po ) (7.33) L = oJE \ PPH

where

we = 360.985647 deg/day is the angular rate of axial rotation of the Earth Po = 1436.068 min = 0.9972696 days is the period of the

geosynchronous orbit

Substituting Eq. (7.33) into Eq. (7.32),

A L = coEn( Pr, H -- Po) (7.34)

The repositioning problem can now be addressed as follows. Given a desired longitudinal shift, say AL = +90 deg, then, from Eq. (7.34),

AL n(Ppn - Po) -- - - -- 0.249317 days (7.35)

O)E

Selecting a value of n allows the solution of (Pr, H - Po). Adding Do solves for PpH. Substitution of Ppn, n, and AL into Eq. (7.32) allows the solution of L. Figure 7.18 is a graph of the A V required to start and stop a longitudinal drift rate


Table 7.1 Reposifioning ofgeosynchronoussatellitesolutionsfor a A L = +90 deg

n rev PPH - Po days Pp~ days nPpH days L deg/day AV m/s

6 0.04155 1.0388 6.233 14.44 82.0 12 0.02078 1.0181 12.217 7.37 41.8 24 0.01039 1.0077 24.184 3.72 21.3 96 0.00260 0.9999 95.987 0.94 5.49

in a geosynchronous orbit as a function of drift rate. Thus, a A V can be associated with a value of L.

Table 7.1 presents a number of solutions to the AL = +90 deg example. Four values ofn were assumed. For each value n, the table presents values of PPH, nPpH (the total elapsed time for repositioning), L, and A V. If the repositioning is to be accomplished in 6 rev, then the drift rate is 14.44 deg/day, and the AV is 82.0 m/s. However, if the repositioning can be done slowly, i.e., in 96 rev, then the drift rate is only 0.94 deg/day, and the AV is only 5.49 m/s. This demonstrates the tradeoff between elapsed time and A V.

The curve on Fig. 7.18 was determined by assuming values of A V, calculating values of the phasing orbit semimajor axis from the energy equation, calculating

1o0 = 421421 km /

~o I

80

70 i /

°o / J ~ so /

,o / / ,o j / '° / / 10

oi// 0 2 4 6 8 10 12 It* 16 18

DRIFT RATE --DEG PER DAY

Fig. 7.18 A V required to start and stop a longitudinal drift rate in a geosynchronous circular orbit.


values of the phasing orbit period from the period equation, and calculating drift rates from Eq. (7.33). The equations and Fig. 7.18 work equally well for westward drifts.

7.2 Terminal Rendezvous

In the final phase of rendezvous before docking, the satellites are in close proximity, and the relative motion of the satellites is all-important. In this phase, it is common to describe the motion of one satellite with respect to the other. In the following subsections, the relative equations of motion will be derived. A solution to these equations will be obtained for the case in which one of the satellites is in a circular orbit.

Derivation of Relative Equations of Motion Figure 7.19 presents the vector positions of the rendezvous and target satellites

at some time with respect to the center of the Earth, r and r r . The position of the rendezvous satellite with respect to the target satellite is p. An orthogonal coordinate frame is attached to the target satellite and moves with it. The y axis is radially outward. The z axis is out of the paper. The x axis completes a fight-hand triad. The angular velocity, a vector of the target satellite, is given by w.

The vector positions of the satellites yield

r = r z + p (7.36)

Differentiating this equation with respect to an inertial coordinate frame results in

i; = r r + ~5 + 2(ua x / ~ ) + cb x p + w x (w x p) (7.37)

Fig. 7.19

rendezvous satellite

i ~ A Y

r T

center of Earth

Geometry and coordinate system for terminal rendezvous.

target satellite


where, as in Eq. (1.2)

}: = the inertial acceleration of the rendezvous satellite i;r -- the inertial acceleration of the target satellite ~5 = the acceleration of the rendezvous satellite relative to

the target satellite 2(w x /5) = the Coriolis acceleration

d~ × p = the Euler acceleration w x (w x p) = the centripetal acceleration

Now, let

i; = g + A (7.38)

where g is the gravitational acceleration and A the acceleration applied by external forces (thrust). Resolving Eqs. (7.37) and (7.38) into the x, y, and z components and solving for the relative accelerations produce

x 2 = - g - + Ax + 2o9~ + (oy + co2x

r

y = - - g ( ~ 7 ~-L) +ay +gr-2092-(ox-.Fo92y (7.39)

Z = - g - + Az

r

Assuming that the target-to-satellite distance is much smaller than the orbit radius of the target satellite or that

/92 • x 2 _4_ y2 _~_ z 2 ~ ( r 2

the following approximate relations can be written

r = [ x 2 + ( y + r r ) 2 + z 2 ] '/2

1- V

X X - g - ~ - - g r - -

F t' T

Z Z - g - ~ - g r - -

r f T

(7.40)

(7.41)


Therefore, the l inearized Eqs. (7.39) become

x 5i = - g r - - + A x + 2o93) + (oy + CO2X

FT

~; = + 2 g r y + A y - 2coJc - (ox + co2y (7.42) ?'T

Z = - g r - - + A z

?'T

When the target is in a circular orbit, & = 0 and co = ~ r ~ ' and Eqs. (7.42)

become

Y = Ax + 2o93)

= A y - 2cok + 3coZy (7.43)

= A z - CO2 z

If there are no external accelerations (e.g., thrust), then,

A x = Ay = A z = 0

Y - 2co3) = 0 (7.44)

y + 2co:~ - 3co2y = 0

Z -'1- COZz = 0

Solution to the Relative Equations of Motion The z equat ion is uncoupled f rom the x and y equat ions and can be solved

separately. Assume a solut ion of the form

Differentiat ing

z = A sin cot + B cos cot (7.45)

= Am cos cot - Bco sin cot

= - A c o 2 sin cot - Bco 2 cos cot

W h e n t = 0, z = zo, and ~ = zo, and so zo = B, and zo = Aco; therefore,

z o . Z --'-- - - sin cot q- Z0 cos cot

co (7.46) = z0 cos cot - z0co sin cot

Subst i tut ion into the ~ equat ion verifies that these equat ions are a solution. In mechanics, they correspond to s imple harmonic motion.

158 O R B I T A L M E C H A N I C S

~t;ATIONS OF MO?IO~

y +zw~ - ~zy. o

~ , e s e rer , d e z v ~ l a e q u a t i o n s a p p l y

~ h e n the t a r g e t i s i n a e l r e u l a r

orbit, I.e. l w - 0 and m -/~ j r~

Ro e x t e r ~ e l f o r c e e a r e c o n s l d e r e d ~

l . e . p A x - Ay = A z - O. ~le

plane I s coincident vlth the

orbit p l a n e o f t h e t a r g e t v e h i c l e .

CtX~DIRATg

R e n d e z v o u s i n g S p a c e c r a f t

Te.r ge t

. /

.v

: e n t e r o f E a r t h

; o

; o

L Fig. 7.20

0

0

6 ( , ~ - s t n w t )

l0 - 3 c o l mt

0

6 . ( 1 - cos ®t)

3m s i n w t

0

0

c o s m%

0

0

h 2 -3t + ~ 81n wt

2 1 ~ s i n ~(- + co, ®t) wt

0 0

-3 + b cos w t 2 s i n

-Z s i n ~ t cos wt

0 0

0

I s t n mt

0

0

x o

Y o

i: Yo

£o

Solution to the first-order circular-orbit rendezvous equations.

The x and y equations are coupled but can be solved to produce

x=xo+2Y°(1-coscot)+ 4 - 6 y o sinot+(6coyo-32o)t 0

y = 4 y o - 2 2o + 2 - 3 y o cos cot + --Y° sin cot 0.) CO

2 = 2yo sin wt + (42o - 6coyo) cos cot + 6coyo - 32o

9 = (3coyo - 22o) sin cot + Yo cos cot

(7.47)

where x0, -~0, Y0, and Y0 are position and velocity components at t = 0. Figure 7.20 presents these solutions in matrix form. This is a compact, descrip-

tive form. Given the initial position and velocity, the position and velocity at some future time can be determined from these equations.

Two-Impulse Rendezvous Maneuver

Given the initial position P0 and velocity P0 for the rendezvous satellite with respect to the target satellite at the origin of the coordinate system and given the desire to rendezvous at a specified time v, the problem is to find A V1 at t = 0 and A V2 at t = T to accomplish rendezvous. Figure 7.21 presents a schematic of this two-impulse rendezvous maneuver.

The solution proceeds as follows. If at time t = 0, the relative position x0, Y0, zo is known (components of P0),

then the relative velocity components -+or, ))Or, Z0r necessary to rendezvous at time


Y

bo

Z

Fig. 7.21 Two-impulse rendezvous maneuver.

ImL X

t = r in the future can be obtained from the x, y, z equations by assuming that x = y = z = 0 and solving for xor, Y0r, Z0r as follows:

ko~ xo sin mr + yo[6~or sin wr - 14(1 - cos ogr)]

09 A

Y0r 2x0(1 -- COS Wr) + y0(4 sin mr -- 3cot cos wr) - - = (7.48) w A

ZOr - -Zo

~o tan wr

where A = 3o~r s inwr -- 8(1 -- cos ~or). The first impulse is given by

A V 1 = [(3¢0r - - 2 0 ) 2 -'I"- (J)0r - - #0) 2 ~- (Z0r - - Z0)2] 1/2 ( 7 . 4 9 )

where ~o, Yo, zo are the actual (initial) velocities of the chaser relative to the target at time t = 0.

The components of the second impulse A V2 are the relative velocities ~ , 2#~, z~ at time t = r , with the initial conditions xo, Yo, zo, and 2Or, Yor, Z0r. Thus,

• 2 - - . 2 x l / 2 AV2 = ( k 2 + y r ~-zr)

The A V2 is necessary to stop the chaser vehicle at the target.

(7.50)

Two-Impulse Rendezvous Maneuver Example Given the AV, 21 m/s, for in-track departure from a circular, synchronous

(24-h) equatorial orbit, this example will investigate the two-impulse rendezvous

160 OHBI IAL MECHANICS

90

8O

. 70

60

50

I 90

80

70

60 / MINIMUM TOT,,L AV ~1.4 m/s

50

40 40

0 2 4 6 8 I0 12 14 16 18 20 22 24

TIME (HOURS)

Fig. 7.22 example.

Total A V vs time from departure to return for the rendezvous maneuver

maneuver to return to the original longitude and orbit in a specified time. This initial A V may be applied in order to avoid some debris. The rendezvous maneuver begins 2 h after the application of the 21-m/s A V and ends at a specified but variable time, as illustrated in Fig. 7.22.

Figure 7.22 presents the sum of the first A V of 21 m/s and the sum of the two impulses required for rendezvous as a function of time from the application of the initial A V to the completion of rendezvous. A minimum value in the curve occurs at 13 h. The minimum total A V is 51.4 m/s. The two-impulse rendezvous maneuver requires 30.4 m/s.

For the minimum AV solution, Fig. 7.23 presents a history of x, in-track, vs y, radial. The position after 2 h is noted. The rendezvous maneuver to return begins at this point and takes 11 h. Figure 7.24 presents ./ vs /9. This figure graphically presents the magnitudes and directions of the A V. The first A V of 21.0 m/s is applied in-track; i.e., 2 = - 2 1 . 0 m/s, and ~ = 0. At t = 2 h, the second AV of 29.0 m/s is applied. Its components are . / = +19.6 m/s and 3' = - 2 1 . 4 m/s. At t = 13 h, the third AV of 1.4 m/s is applied. Its components are x = +1.34 m/s, and 3) = - 0 . 4 0 m/s. Yaw and pitch angles are measured in the x-y plane. Nose up is (+), and nose down is ( - ) . A yaw angle of 0 means that the nose of the satellite is pointed forward, i.e., in the - x direction. A yaw angle of 180 deg means that the nose of the satellite is pointed in the + x direction.

If the satellite can point its engines in any direction, the total A V is the sum of the three A V magnitudes, i.e., 51.42 m/s. However, if the satellite's engines point in the x and y directions but cannot be reoriented, then the total A V is the sum of all the x and y components, i.e., 63.74 m/s, as tabulated on the figure.


RADIAL

SATELLITE

MOTION

t = 2 hours 125-

I 0 0

-125 -I00 -75 -5OA -25

N -25 -

G -INTRAC

E -50-

km

-75 -

- 1 0 0 -

- 1 2 5 -

25 50

RANGE, km

75 i00 125

Fig. 7.23 Relative motion for in-track two-impulse solution; starting at t = 2 h and ending at t = 13 h.

o 15 In

INTRACK p.A CGE RATE, m/5

- I 0

-15

-20

~IANEUyER S EDU_ENCE

AT . = D HOURS, Av I = 21.00 MIS

YAW = O, PITCH - 0

AT ~ - 2 HOURS, ~v 2 = 29.02 M/S

YAW ~ 180% PTTCH = -47?5

k - 19.63 M/S, # - -21.37 M/S

AT ~ -13 HOURS, .~v 3 = 1.40 M/S

YAW = 180", PITCH = -16;2

: 1,34 M/S, '} = -0 .40 M/S

zav ' s (M/S) sav COMPONENTS (M/S)

21. O0 21.00

29, 02 19,63 1.40 21.37

I, 34 51,42 MIS

0.40

63.74 MIS

Fig. 7.24 Relative velocity for in-track two-impulse solution, starting at t = 2 h and ending at t = 13.


7.3 Applications of Rendezvous Equations

Co-elliptic Rendezvous Many space missions require spacecraft rendezvous to dock with another satel-

lite or to perform a rescue or an inspection mission. Typically, a rendezvous mission has a target vehicle in a circular or nearly circular orbit, with the chaser vehicle injected into an orbit of slightly lower altitude. When a specified slant range between the target and chaser vehicle is obtained, the terminal rendezvous phase is initiated. This procedure is often referred to as "co-elliptic rendezvous," for which the conditions

acec = a t e t

(7.51) rpc <rpt

are satisfied, where ac, at, are the chaser and target orbit semimajor axes ec, et =

eccentricity of each orbit, and rpc, rpt a r e the respective perigee radii. A typical rendezvous geometry is shown in Fig. 7.25

The basic advantages of co-elliptic rendezvous are: standardized procedure, choice of lighting conditions, star background, and line-of-sight tracking.

Flyaround Maneuvers After the co-elliptic rendezvous has been performed, a visual inspection of

a satellite in orbit may be desired. This can be performed at shorter or longer distances to the satellite with circular, elliptical, or rectilinear trajectories relative to the target vehicle. The elliptical flyaround trajectory can be achieved over an orbit period by a single radial impulse. The ellipse is in the orbit plane whose

TARGET

f6o KM

MISSION 30 KM •

~osLLipT,C PA~ING ORS~T ~

Fig. 7.25 Typical co-elliptic rendezvous geometry.


Y ~/Yo

¢~v= ;o I 2

¢

Initial position

tot = 90 °

~ , tot = 270 °

_2 2

Fig. 7.26

/tot = 180 °

= xto/y °

. 4

~ t o

) Earth Center

Impulsive flyaround maneuver in orbit plane.

major axis is equal in magnitude to twice its minor axis and is proportional to the magnitude of the applied impulse. The solution is of the form

XCO X -- -- 2(1 - cos cot)

3)0 yco

Y -- -- sin cot 3)o

(7.52)

where 3)o is the radial velocity impulse and co is the orbital angular velocity. A plot of Eq. (7.52) is shown in Fig. 7.26. The relative displacement with respect to the origin of the coordinate reference frame is

/9 = (X 2 + y 2 ) 1 / 2

(7.53) = Y0 [4(1 - cos w t ) 2 + sin 2 cot]l/2

CO

Another approach for close circumnavigation is linear relative translation, which results when the orbit-dependent terms in the rendezvous equations are negligible. The applicable equations are

2 = ax, ~ = ay (7.54)

A circular flyaround at a constant radius can also be performed, which requires a continuous application of thrust to counteract the centrifugal acceleration.


Spacecraft

f

Y

Mass particle

/ ~ O r b i t (circular,

to = orbital r~

Center of Earth

Fig. 7.27 Coordinate system.

Ejected Particle Trajectories Collision with an ejected particle. The possibility of a spacecraft collision

with a particle ejected from a spacecraft is increased greatly if the particle is ejected radially or in a cross-track (out-of-plane) direction. The resulting motion of the particle, to a first-order approximation, is periodic in nature, implying that the particle will return to the ejecting body in an orbit period or a fraction thereof. The in-track (forward or backward) ejection, however, results in a secular increase of the particle distance from the ejecting body, which may or may not be large enough to avoid collision an orbit period later.

Consider the coordinate system attached to a satellite in a circular orbit, as shown in Fig. 7.27, where Ps is the radius of the satellite.

For a particle ejected with a relative velocity k0, y0, or z0 along the x, y, or z axes of the reference frame in Fig. 7.27, the trajectory equations from Fig. 7.20 are of the form

x = ( - 3t + 4sinogt)ko + 2(1-cosogt)j¢o

2 y = -- ( - 1 + cos cot)ko + 3)o sin ogt (7.55)

O9 O9

z = z0 sinogt O9

Solution of Eq. (7.55) leads to the following conclusions:

1) For radial ejection: a) Separation is periodic in time. b) Maximum separation occurs a half-orbit after ejection. c) Separation is reduced to zero upon completion of one orbit.

2) For tangential ejection: a) Separation is always finite and variable with time.


b) In-track separation one orbit after ejection is maximum. c) Separation increases with succeeding orbits.

3) For out-of-plane ejection: a) Out-of-plane ejection periodic in time (change in orbit inclination).

1 and 3 of an orbit period after ejection. b) Maximum separation occurs ~ c) Separation reduced to zero every half-orbit.

For a particle ejected in an arbitrary direction from the satellite, the probability of recontact with the satellite would depend on the magnitudes of the tangential component of velocity.~0. Thus, for example, one period later, i.e., when wt = 27r, the position of the mass (relative frame x, y, z) is given by the equation

6Zrko x - (7.56)

O )

This result shows that the mass will be leading (negative x) the spacecraft for a backward ejection at the initial time t = 0 and lagging for a forward ejection at t = 0 .

A sphere of radius Ps = x can thus be defined as centered at the coordinate frame (spacecraft origin) that will not be entered by the ejected mass one orbit period later if I20l >_ cox/6zr. Consider now a given ejection velocity AV. The x component of A V can be defined as

I~01 : A V c o s ~ (7.57)

where AV is the magnitude of the velocity vector, and fl is a half-cone angle measured from the x axis, as shown in Fig. 7.28.

I I

z

, Y

i i

4

Fig. 7.28 Velocity diagram.

X


The Ix01 > cox/6Jr condition will be satisfied if and only if A V falls within the cone described by fl (either along the positive or negative x axis), and the probabili ty of this occurring can be expressed as

cox ) 2Az P I~ t01>_~- - (7.58)

where

Az ---- an effective area of a spherical zone defined by the cone fl

= 2yr(AV)2(1 - cosf l )

As = an effective spherical area = 4zr(A V) 2

assuming an equal probabil i ty of AV occurring along any direction. Thus, the probabili ty that a mass initially ejected with a velocity A V in an arbitrary direction will be outside a sphere of radius Ps one orbital period following the ejection is

P = 1 - cos fl

li01 - - 1

A V cox (7.59)

= 1 6zrAV

cops --1

6r rAV

The probabili ty that the ejected mass will be within the sphere of radius Ps is then

P p = I - P wps (7.60)

-- 6 r rAV

Thus, for example, for a random ejection from a spacecraft with AV = 10 m/s and Ps = 100 m, the probabil i ty of recontact (collision) in a 500-km circular orbit within a 100 radius is about 5.8 x 10 4.

Debris cloud outline. The linearized rendezvous equations presented in Fig. 7.20 can be used to determine the outline of a debris cloud resulting from a breakup of a satellite in orbit. If, for example, it is assumed that the satellite breaks up isotropically; i.e., the individual particles receive a uniform velocity impulse A V in all directions, then the posit ion of the particles can be computed as a function of time in an Earth-following coordinate frame attached to the center of mass of the exploding satellite to obtain the outline of the resulting cloud. This can be performed as follows:

Consider an explosion or a collision event in a circular orbit such as the one illustrated in Fig. 7.29. An orbiting orthogonal reference frame xyz is centered at the origin of the event at time t = 0 such that x is directed opposite to the


Y AM

t = o

- w

EARTH CENTER

Fig. 7.29 C l o u d d y n a m i c s .

orbital velocity vector, y is directed along the outward radius, and z completes the triad (along the normal to the orbit plane). The linearized rendezvous equations (7.55) can be used to determine the position of a particle leaving the origin of the coordinate frame with a velocity A V; they are of the form

( - ~ 0 4 ) 2 x = + -- sin0 20 + --(1 - cos0)P0

o9 o9

2 Y0 y = - - (cos0 - 1)20 + - - sin0 (7.61)

60 O9

i0 z = - - s in 0

O9

where AV = (22 + p2 + i2)1/2 and 0 = cot. The x, y, z coordinates represent particle position at time t = 0/o9, where 0

is the in-orbit plane angle, and o9 is the angular rate of the circular orbit. The k0, 2~0, i0 terms are initial velocity components imparted to the particle along the x, y, z axes, respectively. It is assumed that A V << V, where V is the orbital velocity of the reference frame.

Equations (7.61) can be normalized with respect to AV/o9 as follows:

x o 9 = ( - 3 0 + 4 sin O)h + 2(1 - cos O)r

AV = X

yo9 = 2(cos0 - 1)h + (sin0)r

AV = Y

Zo9 = n sin 0

AV = Z

(7.62)


>

It > -

12.0

6.0

0.000

- 6 . 0

- 1 2 . 0

0 = 180 °.

= 0 = 2 7 0 °

0 = 4 5 ° = O°

I I t I I I 1 I - 1 6 . 0 - 1 2 . 0 - 8 . 0 - 4 . 0 0 4.0 8.0 12.0 16.0

X = x w l A v

F i g . 7 . 3 0 C l o u d c o n t o u r s i n o r b i t p l a n e .

where

h = Jco/AV, r = ~ o / A V , n = Z o / A V , h 2 -}- r 2 + n 2 = 1

Equations (7.62) can be plotted as a function of 0 for different values of h, r, and n. If, for example, the initial velocity A V distribution for the particles is circular in the x, y plane, then h = cos ~b, r = sin q~, 0 < ~b < 360 deg, and h 2 q- r 2 = 1 with n = 0. The resultant cloud outline is illustrated in Fig. 7.30 for several values of 0.

The accuracy of the results degrades somewhat as time and A V increase compared to the orbital velocity V. The results in Fig. 7.30 are representative of the outline of the debris cloud and can be used to compute the volume of the cloud and the resultant collision hazard to orbiting objects in the vicinity of the cloud, as in Ref. 21, for example.

Acceleration and Velocity Impulse Requirements for a Radial Transfer Trajectory

An orbit-transfer maneuver may be required in which a satellite is transferred from one circular orbit to another along the radius vector. For example, it may be of interest to consider the case in which the orbital transfer is radially outward from an initial offset distance d to a smaller offset distance 6 measured in an Earth-pointing, rotating coordinate frame, as shown in Fig. 7.31.

An approximate solution of the problem can be obtained using the linearized rendezvous equations for circular orbits. In this case, the satellite m at time t = 0 is located at a radial offset distance d and, at a later time, t = T is a radial offset distance 6. The necessary external thrust accelerations ax and ay may be derived and integrated to obtain the total velocity impulses required.


Velocity

t y

I i iz

a y

f

x

a x

--------- ~ = Orbital Rate

~ E a r t h C e n t e r

Fig. 7.31 Radial transfer geometry.

The exact non l inear differential equat ions governing the mot ion of a mass m relative to an Earth-oriented reference frame may be expressed as

3

5~ - 2 c o ~ + x c o 2 rs - 1 = - - L\rf! m

I( 31 ~ + 2w2 + (y + rs)~O 2 r~ - 1 = Ty (7.63) L \ r f / m

~ + ~ o 2 r~ Tz

where

Tx, Ty, T z = external forces

w = circular orbit rate

rs : constant radius of the rotat ing reference frame r f = [x 2 -}- (y + rs) 2 + Z2] 1/2

= radial distance to the mass

x, y, z = negative in-track, radial outward, and out-of-plane

displacement , respectively

I f the ( r s / r f ) 3 term is approximated as ( r s / r f ) 3 ~ 1 - 3y/r~ and only p lanar mot ion is considered, then Eqs. (7.63) become

Y - 2co~ = Tx/m = ax (7.64)

+ 2co2~ - 3co2y = Ty/m = ay


where ax , ay are the negative in-track and radial ly outward accelerations applied to the mass.

For the case of radial transfer, a solution of Eqs. (7.64) can be obtained assuming that x = 2 = 2 = 0 at all t imes and that

ay = ayo = const for 0 < t < tl (7.65)

and

ay = 0 for ta < t < T (7.66)

where T is the transfer time. Thus, for case (7.65), Eqs. (7.64) become

- 2c@ = ax (7.67)

- 3w2y = ay o (7.68)

where Eq. (7.67) represents the Coriolis acceleration resulting from the mass m moving radially outward in a rotating reference frame.

The solution of Eqs. (7.67) and (7.68) is of the form

y = A e nt -[- B e -n t aY° (7.69) n 2

where n 2 = 3co 2 and A, B are constants to be determined. For this case,

= A n e nt - B n e - m (7.70)

= A n 2 e nt + B n 2 e -n t (7.71)

and since, at t = O, y = - d , ~ = O,

which yields

Consequently,

0 = A n - B n

- d = 2 A aY° n 2

lfayo )

( ay° - d ) coshn t aY° y = ~ n 2 _ n 2

- - aY° (cosh ~ / 3 w t - 1) - d cosh ~ / 3 w t 3o) 2

{ aY° - d ) s i n h n t = n \ n 2

( a y o - 3 o ~ 2 d ~ - \ ~/3w ] sinh~/3cot

(7.72)

(7.73)

(7.74)


n2 ( ay° -- d) coshnt Y = \ n 2

= (ay o -- 3o92d) cosh ~/3 wt

The in-track acceleration ax is then

ax = -2o9~ 2

= - ---?g (ay o - 3o)2d) sinh ~/3 cot

(7.75)

(7.76)

Thus, for tl < t < T,

y =

y=

2=

Consequently, for tl 5

-~{exp[n ( t - T)] + e x p [ - n ( t - T)]}

- 8 coshn( t - T)

- 6 n sinhn(t - T)

-3~/3o~ sinla ~/3co(t - T)

--Sn 2 coshn( t - T)

-3co2~ cosh ~/3o~(t - T)

t < T , a y = O a n d

ax = -2~oy

= (2n28/~/3)sinhn(t - T)

The total (combined) acceleration for 0 < t < tl is

a:r = (a 2 + a 2) 1/2

= ( d n 2 - a y o ) 2 s i n h 2 n t + a ~

(7.77)

(7.78)

(7.79)

(7.80)

(7.81)

29(T) = A n e nT - B n e -nT ~ B = A e 2nT

= 0

y ( T ) = A e nT + Ae2nTe -nT

6 = 2 A ( e nT) = - 8 --+ A = - - e -nT

2

For the case tl < t < T when outward radial acceleration (thrusting) is zero, the specific solution of Eqs. (7.67) and (7.68) can be obtained from the conditions.


The velocity impulse (AV) is given by

J0 A V = ardt (7.82)

The actual trajectory of the satellite, as obtained by integrating Eq. (7.63), will always contain an in-track ( - x direction) component. The magnitude of the in-track component can, however, be controlled by varying slightly the radial acceleration component ay o .

7.4 An Exact Analytical Solution for Two-Dimensional Relative Motion

Introduction Interesting and worthwhile solutions for the relative motion of a probe, ejected

into an elliptic orbit in the orbital plane of a space station that is in a circular orbit, are derived by Berreen and Crisp in Ref. 9. They have developed an exact analytical solution by coordinate transformation of the known orbital motions to rotating coordinates.

However, there are three restrictions on the solution of Berreen and Crisp that should be noted: 1) The probe is ejected from the space station at time t = 0 with relative velocity components x~ and y~ but the equations as derived do not permit an initial relative displacement such as P0 = (x 2 + y2)1/2. Generalized equations that permit an initial relative displacement will be derived here. 2) The motion of the probe is restricted to the orbit plane of the space station and is, therefore, two-dimensional. 3) The space station is assumed to be in a circular orbit. As stated previously, restriction 1 will be relaxed in this section by the derivation of orbit element equations for the probe in terms of arbitrary initial relative velocity and displacement components for the probe with respect to the space station. The relaxation of restrictions 2 and 3 will be the subject of future studies.

Geometry and Coordinate Systems Using the notation and description of Berreen and Crisp, 9 consider first the

coordinate systems of Fig. 7.32. The space station-centered system is X, Y; the Xi, Yi system is a geocentric inertial system: and the Xe.Ye system is a geocentric rotating system having its Ye axis always passing through the space station. Coordinates Rp, Op and Rp, ~ are polar coordinates of the probe in the Xi, Yi and X~, Ye systems, respectively.

Uppercase letters are used henceforth for real distances and velocities: and lowercase letters are used for ratios of distance and velocity, respectively, to the station orbital radius Rs and circular orbit velocity Vs. Thus,

x Rp vp X = R s ' rp -- Rs ' vp = Vss (7.83)

where

Vs = ~ (7.84)


v, v,v.

. 1 ~ ~ q , . . .~ '~ lmtmo¢~ r Probe

Fig. 7.32 The rectangular coordinate systems (Xj, Yi),(Xe, Y.), and (X, Y) and the polar coordinates (Rs, 0s), (Rp, 0p), and (Rp, c~) (from Ref. 9).

and/z is the gravitational constant for the Earth. Subscripts s and p refer to station and probe, respectively.

The mean motion N, of the station is

Vs N~ = - - (7.85)

Rs

The angular coordinate Os of the station is then

O,=Nst

with initial conditions defined at t = 0.

(7.86)

Orbital Relations from Berreen and Crisp In an inertial coordinate system, the probe moves in a Keplerian orbit described

by the elements ep, the eccentricity pp, the semilatus rectum ratioed to Rs, and the apsidal orientation 0". Berreen and Crisp 9 describe these elements in terms of the initial relative velocity ratio components x~ and y~ by the equations

pp = (1 - x~) 2 (7.87)

2 = 1 + p p 2 2 ep Vp -- 1 + (1 -- x;)2[yo 2 -F (1 - x ; ) 2 - 2] (7.88) Fp

and

Op = cos- l [ (pp -- 1)/ep] = -- sin-l[(1 -- xo)Yo/ep] (7.89)

with - j r < Op <_ Jr. The next subsection will generalize these equations to include initial relative displacement components xo and yo, as well as x o' and Yo.'

Derivation of Generalized Orbital Relations

Figure 7.33 depicts arbitrary, but still coplanar, initial conditions for the probe and space station.

From the geometry of Fig. 7.33,

and

where

rp = [(l + yo)2 + xg] 1/2 = (l + 2yo + xg + yg) 1/2 (7.90)

vp = [(I- x ; ) 2 = y~2]1/2 = ( 1 - 2x~ + x02 + y~2)1/2

From conservation of angular momentum,

(7.91)

r v (7.92) p p = 2 2 p Phodzontal

Vs~

_ _ !

~)Phorizontal - - 1 - x o

Substituting Eqs. (7.90) and (7.93) into Eq. (7.92).

pp = (1 + 2y0 + x g + yg)(1 - x;) 2


Fig. 7.33

(7.94)

(7.93)

Center of the Eartb

Initial conditions for the probe and space station.


Equation (7.94) is the generalization of Eq. (17) in Berreen and Crisp 9 and reduces to their equation when x0 = Y0 = 0.

From conservation of energy,

2 1 2 __ (7.95) Vp

rp ap

Substituting ap into the eccentricity equation yields

2 2 2 = 1 PP 1 + ppVp (7.96) ep -- - - -~ -- - - ap rp

Substituting Eqs. (7.90) and (7.91) into Eq. (7.96),

2 = 1 _[_ pp I l _ 2x; q_ x; 2 _~_ y : _ 2 1 ep (1 + 2yo + x 2 + yg),/2

(7.97)

Equation (7.94) may be used to substitute for pp. Equation (7.97) is the generalization of Eq. (18) in Berreen and Crisp. 9

The orbit equation is

PP (7.98) rp -- 1 q- ep Cos tip

where tip is the true anomaly. The apsidal orientation Op = -t ip, so that

Op = - c o s -1 (pp/rp) 1 ep

(7.99)

Equations (7.90), (7.94), and (7.97) may be used to substitute for rp, pp, and ep, respectively. Equation (7.99) is the generalization of Eq. (19) in Berreen and Crisp. 9 Thus, exact equations for pp, ep, and 0p have been derived in terms of the initial velocity components x~ and y6 and the initial displacement components x0 and Y0 relative to the space station.

Exact Polar Equations of the Trajectory Berreen and Crisp 9 derive exact equations for the motion of the probe with

respect to the space station in terms of the polar coordinates rp and ot (see Fig. 7.32). The equation for rp is simply the orbit equation expressed in terms of the true anomaly tip or the eccentric anomaly Ep,

PP -- ap(1 + ep c o s Ep) (7.100) rp -- 1 -q- ep c o s f l p


where

Pp ap 1 - e 2 _ p

The variables tip and Ep are related by

(7.101)

tan ~ = t/1--ep'~l/2 tan

\ 1 q - e p ]

Of course, Kepler's equation must be used to relate time t and Ep:

(7.102)

Np(t - t*) = Ep - ep s in Ep (7.103)

where t* is the time of perigee passage for the probe, and Np is the mean motion of the probe given by Kepler's third law,

Ns ~3/2 Np Up

Then, Berreen and Crisp 9 derive an expression for the polar angle y,

(7.104)

Ns y = tip -- ~ p ( E p -- ep sin Ep) (7.105)

where y = o~ - or* and u = or* at t = t*. The equation for y is derived quite directly from

7g = -~ - ( 0 , - O p ) (7.106)

which is evident from Fig. 7.32. Thus, Eqs. (7.100) and (7.105) are exact equations in polar coordinates rp and

g of the probe motion relative to the space station.

Calculation Algorithm for the Exact Solution An algorithm for calculating rp and V is outlined as follows:

1) The radius Rs of the space station orbit is given. The circular orbit velocity Vs is calculated from Eq. (7.84). The initial relative displacement and velocity components of the probe are given. These are normalized by dividing by Rs and

' and y~. Vs to obtain x0, Y0, x0, 2) Calculate pp from Eq. (7.94). 3) Calculate ep from Eq. (7.97). 4) Calculate ap from Eq. (7.101). 5) Calculate Ns/Np from Eq. (7.104). 6) Calculate rp at t = 0 from Eq. (7.90). 7) Calculate Ep at t ---- 0 f r o m Eq. (7.100).


8) Calculate tip at t = 0 from Eq. (7.102). 9) Calculate g at t = 0 from Eq. (7.105).

Then, for other times, 10) Calculate Ep from Eq. (7.03), where Ns is calculated from Eq. (7.85) and

Np from Eq. (7.104). 11) Calculate tip from Eq. (7.102). 12) Calculate rp from Eq. (7.100). 13) Calculate ~, from Eq. (7.05).

7.5 Optimal Multiple-Impulse Rendezvous

Two-Impulse Time-Fixed Rendezvous Two-impulse time-fixed rendezvous between satellites in neighboring orbits was

investigated in Sec. 7.2. Linearized relative equations of motion were solved for given boundary conditions and a specified time to obtain the two vector impulses required. No optimization was involved; although, in the example headed "Two- Impulse Rendezvous Maneuver" in Sec. 7.2, the time for rendezvous was varied so that a tradeoff of total A V vs rendezvous time could be developed. A solution for a minimum total A V was identified in the tradeoff. Succeeding sections will identify optimal solutions involving two, three, and four impulses.

Optimal Multiple-Impulse Rendezvous Between Satellites in Neighboring Orbits

A landmark development in optimal multiple-impulse rendezvous was the doc- toral thesis of John Prussing in 1967.1° Prussing considered neighboring nearly circular orbits. He applied Lawden's theory of the primer vector to the equations of motion, linearized about an intermediate circular orbit. Descriptions of the primer locus are used to develop two-, three-, and four-impulse optimal solutions. Figure 7.34 is Fig. 9.4 of Ref. 10. It presents AV/SR vs tF for fl/6R = 81.4 deg, where

A V ---- the sum of the impulses R = the nondimensional difference between the final and initial circular

orbit radii tF = time measured in reference orbit periods

fl = flH + Aft = 0H + A0 of Fig. 7.1

Figure 7.34 shows that two impulses are optimal for small values of tF. As tF increases, three- and four-impulse solutions become optimal. The indicator 3 + means that a three-impulse solution with a final coast is optimal. As time increases, the Hohmann A V cost is finally achieved.

Fig. 7.35, which is Fig. 9.1 of Ref. 10, presents final state variations for circle-to- circle coplanar rendezvous, which are reached optimally using different numbers of impulses, where

3 (~OF = t~ - - -3 Rtv (7.107)

4


&V

2.0

1.8 / ~R = l '42 RADIANS (81'4°)

1.6

1.4

1.2: • I 2:-IMPULSE

0 . 8 - -

0 . 6 - -

I I I o.4 I I I

I , I I I = + 0 l 2 ~ ~ - - - - ~ 2: ~-i = 3 3 F

I J I o i i

0.5 1.0

Fig. 7 .34 Ref . 10).

i ~ l - -~ - HOHMANN COST

I i I I

- - - - 1 4 1 - - 3

I I tr

L5 2:.0

O p t i m a l c o s t as a f u n c t i o n o f t r a n s f e r t i m e for g i v e n in i t ia l c o n d i t i o n s ( f r o m

Figure 7.35 summarizes the results of Ref. 10. These results are also described in Refs. 11 and 12.

Gobetz and Doll 13 provide an excellent survey article of orbit transfers and rendezvous maneuvers. They describe the problems and solutions, and present graphic results and a bibliography of 316 papers, articles, and reports.

Optimal Multiple-Impulse Nonlinear Orbital Rendezvous Reference 14 extends the circle-to-circle solutions of the preceding subsection

to the nonlinear case, i.e., in which the difference in orbit radii may be large. Again, primer vector theory is used to obtain the optimal number of impulses, their times and positions, and the presence of initial or final coasting arcs. Reference 14 is the journal article version of Ref. 15.

Figure 7.36 which is Fig. 4 of Ref. 14, presents total AV vs time in reference orbit periods for a final/initial-orbit radius ratio of 1.6 and for fl = 0, 90, 180, 270 deg. The A V curves all decrease with time and eventually reach the Hohmann A V cost. The optimal number of impulses are designated at various points on these curves.

An interesting example presented in Refs. 14 and 15 is the optimal trajectory for rendezvous with a target satellite in the same circular orbit at a phase angle fl of 180 deg in a specified time of 2.3 reference orbit periods. Figure 7.37, which is Fig. 5.14 of Ref. 15, depicts the trajectory, the locations of the four optimal impulses, and a tabular listing of the A V magnitudes and their application times. The total cost is A VT/Vc = 0.189. By comparison, the cost of the best two-impulse rendezvous is AV~/Vc = 0.224. This is the cost that would be obtained by the technique described in the subsection headed "In-Orbit Repositioning" in Sec. 7.1.

R E L A T I V E M O T I O N IN O R B I T 179

1077

871"

6T/"

4 T/" ~-

27'/"

Fig. 7.35 Ref. 10).

/ BeF /

i / / /

I ,"1 / / ~ \ / / /

I # . .~/ ~ \ ...i"1

(c: Av ~ 4 ~" ~ - - - - ' k " /

/ ~ ~ ~ , ~ ' ~ ' ~ ' ~ MOHMANN {C :O .5 ) / ? " F ' - ~ i I

0.5 1.5 2.O

t F

1,0 + REFERENCE ORBIT PERIODS

Reachable final state variations; optimal multiple-impulse solutions (from

1.5 | 3 1 I I I

2 \ : - - \ . ~ ~. .~ ~ '~ . : - -_ .~ ~

>

<3

1 . 0 - -

0 . 5 -

0.0

m m ] ~ = 0

- - . m ] ~ = 9 0 °

. . . . B = 1 8 o -

~ m ~ = 270 ~ Hohmann Cost

] I I l 0.5 1.0 1.5 2.0

Time Fig. 7.36 A V vs time plot for R = 1.6 (from Ref. 14).

1 2 5


t . O

0.5

~ - 0 . 0

- .5

- ! .0

..... :::::::::::::::::::::::::::::::::::::::::::::::: ........... / : / ' . . . - - ~ - ' ~ - ~ '..% "..

..... / \'-;:..... •

( / ~ .,.

/! / i i! t . .1,

k I/ :, ,\ / ........ ::, \ /:: .....

,;.: .....,..:5~ ~ ./...: ...... ........ "--.: ...-:~..~ _......21/. .......

Fig. 7.37

' Rr'= 1.0 'Tr=2.3' /3 =180 ~=0.0

o 67==0.0072 l~=O.O000 o &Vl .0.0274 1a.0,8027 • &V==0.0233 l=-1,533g '~ 8V4-0.0714 I ,o2.3000

AV~=O. I893 . . . . . . . . T r o n s f e r T r o i e c t o r y

-1 .5 -1 .5 -11.0 -15 0.0 I 015 110 115 2.0 = 215 3.0 = 3,3

Example rendezvous trajectory with four impulses (from Ref. 15).

Other Optimal Multiple-Burn Rendezvous-Type Maneuvers Other optimal rendezvous-type maneuvers have been developed for important

orbital applications. Some of these are briefly summarized here chronologically. The paper by Holaday and Swain 16 entitled "Minimum-Time Rescue Trajec-

tories Between Spacecraft in Circular Orbits" uses the nonlinear equations of motion. The multipoint boundary-value problem was solved using the indirect shooting technique with a modified Newton's method for convergence. All of the illustrated minimum-time trajectories describe thrust-coast-thrust histories. If sufficient fuel is available, the optimal solution may utilize continuous maximum thrust. The effects of an altitude constraint are examined. The effects of phase angle, specific impulse, and maximum thrust on rendezvous time are displayed.

Optimal Impulsive Time-Fixed Direct-Ascent Interception Prussing and Clifton ]7 obtained minimum-fuel, impulsive solutions for the prob-

lems of attack avoidance on a satellite, followed by a return to the original orbit station. The evasion distance and time are constrained. Both constrained and free final-time cases are considered. Primer vector theory is used to obtain the optimal solutions presented, which include three-impulse solutions for an arbitrarily specified evasion radius vector and two-impulse free-return trajectories for certain evasion radius vectors. Primer vector histories are displayed for a number of examples.

Optimal Trajectories for Time-Constrained Rendezvous Between Arbitrary Conic Orbits

Wellnitz and Prussing TM generate optimal impulsive trajectories for time-constrained rendezvous between arbitrary conic orbits. Primer vector theory is used to


determine how the cost in A V can be minimized by the addition of initial and final costs and by the addition of midcourse impulses.

A universal variable formulation was used. Results are presented for a rendezvous between unaligned coplanar elliptical orbits and for coplanar and inclined elliptical rescue missions.

Optimal Cooperative Time-Fixed Impulsive Rendezvous In Ref. 19, Mirfakhrale, Conway, and Prussing developed a method for de-

termining minimum-fuel trajectories of two satellites. The method assumes that the satellites perform a total of three impulsive maneuvers, with each satellite being active, i.e., performing at least one maneuver. The method utilizes primer vector theory and analytical expressions for the gradient of the total amount of fuel expended. Results are presented for a number of cases and demonstrate the advantage of performing a rendezvous cooperatively for certain initial geometries and times of flight.

Optimal Orbital Rendezvous Using High and Low Thrust In Ref. 20, Larson, and Prussing use optimal-control theory to examine a spe-

cific class of satellite trajectory problems where high- and low-thrust propulsion systems are used. These problems assume that a satellite is in an established orbit about a planet. An intercept of a predetermined position in space in a specified amount of time using an optimal high-thrust program is then executed. Finally, the satellite returns to the original orbit station using the low-thrust propulsion system in an optimal fashion. Solutions are obtained for problems with a fixed final time. Results for two examples are presented.

References

IBillik, B. H., and Roth, H. L., "Studies Relative to Rendezvous Between Circular Orbits," Astronautica Acta, Vol. 12, Jan.-Feb. 1967, pp. 23-26.

2pitkin, E. T., "A General Solution of the Lambert Problem," The Journal of the Astronautical Sciences, Vol. 15, 1968, pp. 270-271.

3Lancaster, E. R., and Blanchard, R. C., "A Unified Form of Lambert's Theorem," NASA Technical Note D-5368, 1969.

4Herrick, S., Astrodynamics, Vol. 1, Van Nostrand, 1971. 5Battin, R. H., An Introduction to the Mathematics and Methods of Astrodynamics,

AIAA, New York, 1987. 6Gooding, R. H., "A Procedure for the Solution of Lambert's Orbital Boundary-Value

Problem," Celestial Mechanics and Dynamical Astronomy, 48, 1990, pp. 145-165. 7prussing, J. E., and Conway, B. A., Orbital Mechanics, Oxford University Press, 1993. SHanson, J. T., "Optimal Maneuvers of Orbital Transfer Vehicles," Ph.D. Dissertation,

Univ. of Michigan, Ann Arbor, MI, 1983. 9Berreen, T. E, and Crisp, J. D. C., "An Exact and a New First-Order Solution for

the Relative Trajectories of a Probe Ejected from a Space Station," Celestial Mechanics, Vol. 13, 1976, pp. 75-88.

l°Prussing, J. E., "Optimal Multiple-Impulse Orbital Rendezvous," Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1967.


11Prussing, J. E., "Optimal Four-Impulse Fixed-Time Rendezvous in the Vicinity of a Circular Orbit," AIAA Journal, Vol. 7, May 1969, pp. 928-935.

12prussing, J. E., "Optimal Two- and Three-Impulse Fixed-Time Rendezvous in the Vicinity of a Circular Orbit," AIAA Journal, Vol. 8, July 1970, pp. 1221-1228.

13Gobetz, E W., and Doll, J. R., "A Survey of Impulsive Trajectories," AIAA Journal, Vol. 7, May 1969, pp, 801-834.

14prussing, J. E., and Chiu, J.-H., "Optimal Multiple-Impulse Time-Fixed Rendezvous Between Circular Orbits," Journal of Guidance, Control, and Dynamics, Vol. 9, Jan.-Feb. 1986, pp. 17-22.

15Chiu, J.-H., "Optimal Multiple-Impulse Nonlinear Orbital Rendezvous," , Ph.D. Thesis, Univ. of Illinois at Urbana-Champaign, IL, 1984.

16Holaday, B. H., and Swain, R. L., "Minimum-Time Rescue Trajectories Between Spacecraft in Circular Orbits," Journal of Spacecraft and Rockets, Vol. 13, July 1976, pp. 393-399.

17prussing, J. E., and Clifton, R. S., "Optimal Multiple-Impulse Satellite Avoidance Maneuvers," AAS Paper 87-543, Aug. 1987.

18Wellnitz, L. J., and Prussing, J. E., "Optimal Trajectories for Time-Constrained Rendezvous Between Arbitrary Conic Orbits," AAS Paper 87-539, Aug. 1987.

19Mirfakhrale, K., Conway, B. A., and Prussing, J. E., "Optimal Cooperative Time-Fixed Impulsive Rendezvous," AIAA Paper 88-4279-CE Aug. 1988.

2°Larson, C. A., and Prussing, J. E., "Optimal Orbital Rendezvous Using High and Low Thrust," AAS Paper 89-354, Aug. 1989.

21Chobotov, V. A., "Dynamics of Orbital Debris Clouds and the Resulting Collision Hazard to Spacecraft," Journal of the British Interplanetary Society, Vol. 43, May 1990, pp. 187-194.

Problems 7.1. Assume an Earth satellite in circular orbit, altitude = 278 km and period --- 90 min. Assume two-body motion, i.e., no atmospheric drag, no sun, moon perturbations, etc.

At an arbitrary time, a small free-flying experiment package is ejected with a small A V from the satellite. Formulate the equations of relative motion for X/A V and y / A V for two cases: 1) the A V is applied in the direction of satellite motion; and 2) the AV is applied radially outward. Then, plot Y / A V vs X/AVfor one period of satellite motion (90 min) for both cases. How far will the free flyer be from the satellite after 90 min?

7.2. A space station is in a 90-min period circular orbit around a spherical, atmo- sphereless Earth. At t = 0, a nearby remote telescope has the following relative position and velocity components in a rendezvous-type coordinate system whose origin is on the space station x0 = 0, Y0 = 13500/zr m, z0 = 0, 2o = 10 raps, Y0 = 0, z0 = 0. How far, in meters, is the telescope from the space station 15 rain later? What is the magnitude of the relative velocity, in mps, at this time?

7.3. Buck Rogers in a space bug and Dr. Huer in a space station are in the same circular orbit (P = 2 h) about the Earth but are 5486 m apart. Since Buck is ahead of the station and wants to rendezvous with his old friend in 30 min, he decides to


aim his rockets at the station and change his velocity by 3.05 mps (retrofire). After ingeniously determining Buck's trajectory, Dr. Huer exclaims, "Buck's goodfed again!" But has he? To find out, write down the x, y, 2, and ~ equations of Buck's motion relative to Dr. Huer. Determine the distance (in meters) between the two and their relative velocity (in meters/second) after 30 rain. At this time, is Buck moving toward or away from the space station?

7.4. In order to avoid some orbiting debris, a geosynchronous satellite in a circular equatorial orbit applies an in-track A V. Some time later, the debris has passed, and the satellite has the following relatived position and velocity with respect to its nonmaneuvering location: x = -120.6 km, y = 71.12 km, 2 = -10.636 m/s, and ~ = 20.221 m/s. The satellite will now initiate a two-impulse maneuver to return to its original location in 2 h. What are the magnitudes and directions of A V1 and A V2 ?

7.5. "Phooey !" exclaims Buck Rogers as he throws his sandwich out the airlock of the satellite which he shares with Dr. Huer. "You've goofed again, Buck!" chides Doc. "You know that we're in a circular orbit about the moon. Since you threw that sandwitch radially away from the center of the moon, we'll have tuna salad all over our portholes after one revolution (3 h)." After a lightning calculation, Buck chortles, "You're wrong this time, Doc. You used the first-order rendezvous equations. I used the exact solution of Berreen and Crisp. Assuming a relative A V of 0.01 Vcirc~la~, the tuna salad will miss us by at least 2000 m." What will the distance between the sandwich and the satellite be after one revolution? Use the exact polar equations of Berreen and Crisp and the calculation algorithm presented in Sec. 7.4.

7.6. Use the equations for in-orbit repositioning to shift the longitude of a satellite in a geosynchronous circular equatorial orbit by +12 deg in 3 revolutions of the phasing orbit. Calculate the maneuver A VI, to start the longitude drift. After 3 revolutions the application of AV2 = -AV1 will stop the drift. Compare the position of the satellite at this time with the position determined from the solution to the relative equations of motion, Eqs. (7.47), when A V1 is applied.

7.7. Solve for relative range and magnitude of velocity between a free-flying space object (e.g. telescope) and a space station in a circular orbit. Use equations in Fig. 7.20 for any specified initial conditions of the space object.

7.1.

Selected Solutions x 4

Case 1: = - - - sin cot + 3t AV co

Y 2 - - - ( 1 - c o s c o t ) AV co

After 90 rain, ~v = 16,200 s


X 2 Case 2: -- ( 1 - c o s w t )

AV co

Y 1 - - -- sin cot AV co

After 90 min, 0

7.2. Relative distance = 7746.86 m Relative velocity = 6.61 m/s

7.3. Relative distance = 10,600 m Relative velocity = 11.0 m/s Moving away

7.4. AVI = 39 .30m/s AV2 = 19.08 ngs

7.5. Orbit radius = 2.437 x 106 m Distance = 2297 m (Buck is right)

7,6, AV1 = l l . 2 6 m / s Position difference -- 3390 m or 0.0046 deg

7,7, A solution to this problem is provided in the software that accompanies this book.