Acta Astronautica Vol. 50, No. 1, pp. 49–53, 2002? 2002 Published by Elsevier Science Ltd.

Printed in Great Britainwww.elsevier.com/locate/actaastro PII: S0094-5765(01)00140-0 0094-5765/02/$ - see front matter

ACADEMY TRANSACTIONS NOTE

ORBITAL INSPECTION VEHICLE TRAJECTORIES BASED ONLINE-OF-SIGHT MANEUVERS

TREVOR WILLIAMS†; ‡Department of Aerospace Engineering, University of Cincinnati, P.O. Box 210070 Cincinnati,

Ohio 45221-0070, USA

(Endorsed by Prof : Peter Bainum§; Received 2 April 2001)

Abstract—This Note derives techniques for automatically generating trajectories that are wellsuited to implementation by simple orbital inspection vehicles: the vehicle leaves a speciCed point,coasts until it crosses a given switching line, applies a Dv of given magnitude and direction, andthen coasts until it reaches its target position. ? 2002 Published by Elsevier Science Ltd.

1. INTRODUCTION

There is currently considerable interest in the de-velopment of remotely piloted vehicles capable ofperforming Ey-arounds of the International SpaceStation on-orbit, to inspect for damage or moni-tor extravehicular operations. NASA Eight-testedthe Autonomous EVA Robotic Camera (AERCam)Sprint vehicle [1], the prototype of a family ofspacecraft envisaged for International Space Sta-tion inspection tasks, during the STS-87 mission inDec. 1997. Two weeks later, the Daimler Inspectorfree-Eyer [2] was released from a Progress resup-ply craft, with the planned mission (not actuallyaccomplished) of performing a Ey-around of Mir.These vehicles were both Eown under remote con-trol by a human operator, essentially by means ofvisual cues alone, a type of operation that is likelyto also be used for early operational inspectionvehicles.

This Note examines a class of trajectory that iswell suited to implementation by simple orbital in-spection vehicles. In these transfers, the vehicleleaves a speciCed point and coasts until it crossesa given switching line, deCned by the alignmentof particular visual cues on the space station. Atthis point, an impulsive Dv of given magnitudeand direction is applied, representing a burn of

†Tel.: +1-513-556-3221; fax: +1-513-556-5038.E-mail address: trevor.williams@uc.edu (T. Williams).‡Associate Professor. Associate Fellow AIAA.§IAA Member, Section 2.

speciCed duration and orientation: such a ma-neuver can readily be accomplished by a vehi-cle such as AERCam Sprint, which is equippedwith attitude hold and thruster pulsewidth con-trol capabilities. The vehicle then coasts untilit reaches the desired target position. This typeof transfer allows orbital mechanics to be ex-ploited, so reducing propellant consumption ascompared with “brute force” methods, withoutrequiring the availability of full range and rangerate data. Two examples of such trajectories,for diNerent starting positions “o”, are shown inFig. 1, taken from [3]. These transfers representEight around the tail of the Space Shuttle Orbiter,Eying in gravity-gradient attitude (tail down, bellyforward) at 185 naut mi, performed using theSAFER astronaut maneuvering unit; the switch-ing line corresponds to those positions at whichthe eyeline of the astronaut is aligned with theunderside of the Orbiter.

It should be noted that the Cgures presented in[3] were obtained using a time-consuming iter-ative technique. The new results derived in thisNote allow trajectories of this type to be generatedautomatically for given initial and Cnal positions,switching line geometry, and Dv magnitude anddirection. The analysis is based on the Clohessy–Wiltshire (CW) equations [4] that describe the rel-ative motion of two spacecraft in close orbits. Theclosed-form solutions [5] to these equations formthe basis for the well-known CW targeting tech-nique, whereby the initial velocity can be found thatwill set up a coast between two speciCed points in

49

50 T. Williams

Fig. 1. Two transfers, each with 0:2 fps line-of-sight maneuver.

a speciCed time; this approach was also extendedto low-thrust transfers in [6]. However, the geom-etry of coasting transfers is too restricted to allowmany of the missions that will be required of opera-tional inspection vehicles. Switching line Dv trans-fers are much more general, as well as being lesssensitive to initial velocity errors, as a result of the“pilot-in-the-loop” feedback nature of the timing ofthe maneuver [3]. However, their analysis is alsomuch more complicated than for traditional CWtargeting. The derivations that are required for thisanalysis are given here, providing a complete char-acterization of the desired trajectories. These resultsalso yield insight into the existence and uniquenessof the new class of transfer.

2. PROBLEM FORMULATION

The CW equations that govern relative motion inthe plane of a circular reference orbit of angularrate n are given as

Px − 2nz= ax; (1a)

Pz − 3n2z + 2nx= az (1b)

in the local vertical=local horizontal (LVLH) coor-dinate system that is commonly used for on-orbitproximity operations: the LVLH x-axis is directedalong the velocity vector (or VBAR) of the refer-ence satellite, and the z-axis along its downwardradius vector (or RBAR). For coasting Eight, i.e.between maneuvers, the applied accelerations axand az are identically zero. The equations then haveclosed-form solutions [5] which are given in termsof the initial relative position (x0; z0) and velocity

(x0; z0) of the maneuvering satellite as

x(t) = −2(z0=n) cos nt + [4(x0=n) − 6z0] sin nt

+[6z0 − 3(x0=n)]nt + [x0 + 2(z0=n)];(2a)

z(t) = [2(x0=n) − 3z0] cos nt + (z0=n) sin nt

+[4z0 − 2(x0=n)]: (2b)

DeCning � ≡ nt, the angle travelled by the refer-ence satellite around its orbit since time t= 0, andthe new variables u0 ≡ x0=n and v0 ≡ z0=n (whichhave dimensions of distance), these solutions canbe rewritten as

x(�) = x0 + 6z0(�− sin �) + u0(4 sin �− 3�)

+2v0(1 − cos �); (3a)

z(�) = z0(4 − 3 cos �) + 2u0(cos �− 1)

+ v0 sin �: (3b)

The type of trajectory that will be examined in-volves Eight from a speciCed initial position (x0; z0)to a speciCed Cnal position (x2; z2). The transfer in-volves coasting Eight with the exception of an im-pulsive maneuver, with pre-speciCed componentsDvx and Dvz, that is applied when the trajectoryintersects a given line, described by the equation

z= �x + �: (4)

This type of maneuver is one that is feasible for arelatively simple close-range orbital inspection ve-hicle: a burn of speciCed duration and orientation isapplied when the spacecraft becomes aligned with

Inspection vehicle trajectories 51

some given visual features on the main spacecraft.Note that the speciCc point (x1; z1) at which themaneuver is applied on the switching line is notspeciCed a priori; neither are the Eight times t1to the maneuver and t2 from it to the target. Ifthese leg times, as well as the initial scaled velocitycomponents u0 and v0, can be found, then all otherquantities, e.g. x1 and z1, can be calculated fromthem, so describing the transfer completely.

3. TRAJECTORY CHARACTERIZATION

The maneuver point (x1; z1) lies on the switchingline, and so must satisfy z1−�x1 = �. But x1 and z1also correspond to the CW solution eqns. (3) prop-agated to scaled time �1 = nt1. Combining theseexpressions allows the unknown v0 to be given interms of u0 and �1 as

v0 = [f(�1)u0 + g(�1)]=h(�1); (5)

where

f(�1) = �(4 sin �1 − 3�1) + 2(1 − cos �1); (6a)

g(�1) = � + �x0 + z0[6�(�1 − sin �1)

+3 cos �1 − 4]; (6b)

h(�1) = 2�(cos �1 − 1) + sin �1: (6c)

Substituting eqn. (5) back into eqns. (3), evaluatedat �= �1, then gives the maneuver position com-ponents in terms of u0 and �1 alone as

x1 = [F(�1)u0 + G(�1)]=h(�1); (7a)

z1 = [�F(�1)u0 + H (�1)]=h(�1); (7b)

where

F(�1) = (4 sin �1 − 3�1)h(�1)

+2(1 − cos �1)f(�1); (8a)

G(�1) = [x0 + 6z0(�1 − sin �1)]h(�1)

+2(1 − cos �1)g(�1); (8b)

H (�1) = �G(�1) + �h(�1): (8c)

Making use of eqn. (5) in a similar fashion to elim-inate v0 from the time derivative of eqns. (3), againevaluated at �1, gives the relative velocity compo-nents immediately prior to application of the Dv as

x1 = n[A(�1)u0 + B(�1)]=h(�1); (9a)

z1 = n[C(�1)u0 + D(�1)]=h(�1); (9b)

where

A(�1) = (4 cos �1 − 3)h(�1) + 2 sin �1f(�1);(10a)

B(�1) = 6z0(1 − cos �1)h(�1) + 2 sin �1g(�1);(10b)

C(�1) = − 2 sin �1h(�1) + cos �1f(�1); (10c)

D(�1) = 3z0 sin �1h(�1) + cos �1g(�1): (10d)

If we deCne the new scaled velocity variables u1 ≡x1+=n and v1 ≡ z1+=n, where x1+ = x1 + Dvx andz1+ = z1 +Dvz are the relative velocity componentsimmediately following the maneuver, then eqn. (9)clearly implies that

u1 = [A(�1)u0 + SB(�1)]=h(�1); (11a)

v1 = [C(�1)u0 + SD(�1)]=h(�1); (11b)

where

SB(�1) =B(�1) + (Dvx=n)h(�1); (12a)

SD(�1) =D(�1) + (Dvz=n)h(�1): (12b)

Now, propagating eqns. (3) through scaled time�2 = nt2 from initial conditions corresponding tothe end of the maneuver (relative position (x1; z1)and scaled velocity (u1; v1)) must yield the desiredCnal position (x2; z2). Using eqns. (7) and (11)to express the post-maneuver initial conditions interms of the two unknowns u0 and �1, this propa-gation can be shown to yield, after some algebra,

x2 = [P(�1; �2)u0 + Q(�1; �2)]=h(�1); (13a)

z2 = [R(�1; �2)u0 + S(�1; �2)]=h(�1); (13b)

where

P(�1; �2) = (4 sin �2−3�2)A(�1)

+2(1−cos �2)C(�1)

+[1+6�(�2−sin �2)]F(�1);(14a)

Q(�1; �2) = (4 sin �2−3�2) SB(�1)

+2(1−cos �2) SD(�1)

+G(�1)+6(�2−sin �2)H (�1);(14b)

R(�1; �2) = 2(cos �2−1)A(�1)+sin �2C(�1)

+ �(4−3 cos �2)F(�1); (14c)

S(�1; �2) = 2(cos �2−1) SB(�1)+sin �2 SD(�1)

+ (4−3 cos �2)H (�1): (14d)

52 T. Williams

Fig. 2. Characteristic function vs. t2; 0:4 fps line-of-sight maneuver.

Fig. 3. Solution time pairs (t1; t2); 0:4 fps line-of-sight maneuver.

Rewriting eqns. (13) gives two alternative expres-sions for the initial scaled velocity component u0

in terms of the target position components:

u0 = [h(�1)x2 − Q(�1; �2)]=P(�1; �2)

= [h(�1)z2 − S(�1; �2)]=R(�1; �2): (15)

Since both of these expressions must hold for thedesired solution trajectory, they can be rearrangedto give its characteristic equation:

W (�1; �2) ≡ R(�1; �2)[h(�1)x2 − Q(�1; �2)]

−P(�1; �2)[h(�1)z2 − S(�1; �2)] = 0:(16)

The roots of this equation consist of all possiblesolution transfer time pairs (�1; �2). The scaled ini-tial velocity component u0 corresponding to eachsolution can then be computed directly from ei-ther version of eqn. (15), v0 from eqn. (5), and x1

and z1 from eqns. (7), thus completely describingthe transfers. This procedure can be regarded asthe (more complicated) equivalent, for switchingline Dv trajectories, of traditional CW targeting forcoasting transfers.

4. PROPERTIES OF SOLUTION TRAJECTORIES

The properties of the characteristic functionW (�1; �2) will now be investigated and conclu-

Inspection vehicle trajectories 53

sions drawn concerning the associated trajectories,with numerical results given for illustration.

One point to note Crst is that the orbital rate nonly enters into the derivation ofW (�1; �2) througheqns. (12). Thus, if the maneuver is representedin terms of scaled velocity variables �u ≡ Dvx=nand �v ≡ Dvz=n, W (�1; �2) is independent of n.Consequently, a transfer that is a solution for somegiven orbital altitude will also be valid at any otheraltitude so long as �u and �v are equal in bothcases, i.e. so long as the actual Dv is scaled by theappropriate value for n.

A question of considerable interest is that of theexistence and uniqueness of solutions to the char-acteristic equation W (�1; �2) = 0. In order to studythe distribution of solution pairs (�1; �2), it is con-venient to Cx �1 and examine how W (�1; �2) be-haves as a function of �2 alone; doing this for all �1

values of interest then provides a complete descrip-tion. Expanding eqn. (16) and simplifying, makinguse of eqns. (14), shows that W (�1; �2) then hasthe form

W (�1; �2) = ca + cb sin(�2 + ’b)

+ cc�2 sin(�2 + ’c); (17)

the coeTcients ca; cb and cc depend on �1. For largevalues of �2, the Cnal term will dominate: solu-tion values for �2 will then occur that are sepa-rated by roughly ! radians, corresponding to leg 2transfer times that diNer by roughly half an orbitalperiod.

However, the transfers that are of most practi-cal interest are those for which both leg times arerelatively short. The distribution of solutions inthis region is more complicated, as all three termsin eqn. (17) may be signiCcant. Figure 2 showsthe type of behavior that can result: this plots W ,corresponding to Space Shuttle Ey-around trajec-tories (with x0 = − 11 ft) of the form shown inFig. 1, as a function of t2 for four Cxed t1 values.It can be seen that two solutions exist in the ranget26 15 min for t1 values of 6 and 15 min; how-ever, for t1 equal to 9 or 12 min, no solutions lie in

this range. Figure 3 provides more detail by plot-ting all solution pairs in the region of interest: thisreveals that no short-duration solution exists for7:4 min¡t1 ¡ 13:7 min. Outside this range, botha shorter (more direct) and a longer (more curvi-linear: the outside transfer in Fig. 1 is an example)duration solution typically exists, with these coa-lescing at the limiting t1 values (the inside transferin Fig. 1 corresponds to such a coalescence).

5. CONCLUSIONS

This Note has derived techniques for automaticallygenerating switching line Dv trajectories, in whichthe vehicle leaves a speciCed point, coasts until itcrosses a given switching line, then applies a Dvof given magnitude and direction and coasts untilit reaches its target position. Conclusions were alsoreached concerning the existence and uniquenessof such transfers.

REFERENCES

1. Williams, T. W. and Tanygin, S., On-orbitengineering tests of the AERCam Sprint roboticcamera vehicle. Paper AAS-98-171, Proceedingsof the Eighth AAS=AIAA Space�ight MechanicsMeeting; Monterey, CA, February 1998.

2. Kaschner, J., The X-Mir-Inspector mission. PaperISTS 98-c-16, 21st International Symposium onSpace Technology and Science, Omiya, Japan, May1998.

3. Williams, T. W. and Baughman, D., Exploitingorbital eNects for short-range extravehiculartransfers. Paper AAS-93-181, Proceedings of theThird AAS=AIAA Space�ight Mechanics Meeting(Advances in the Astronautical Sciences; vol. 82;Part II); Pasadena, CA, February 1993, pp. 1201–1214.

4. Clohessy, W. H. and Wiltshire, R. S., Terminalguidance system for satellite rendezvous. Journal ofthe Aerospace Sciences, 1960, 27, 653–658.

5. Kaplan, M. H., Modern Spacecraft Dynamics andControl. Wiley, New York, 1976.

6. Williams, T. W. and Tanygin, S., Propulsionsystem sizing for orbital inspection vehicles. Journalof Guidance; Control; and Dynamics, 1999, 22,375–378.