Control and stability problems of remote orbital capture

Mechl;qism and Machine Thanty, 1977, Vol. 12, pp. 57-64. Per(gtmon Preu. Pflntld in Great Britain

Control and Stability Problems of Remote Orbital Capturei" M. H. Ksplsn¢

and A. A. Nadksrni•

Received 9 June 1975 Abstrsct Certain space shuttle missions may require retrieval of passive spinning and precessing satellites. One proposed means of .retrieval utilizes a free-flying teleoperator launched from the shuttle. A study of misalignment, stability, and certain control aspects during capture of an object is reported here. The approach used is to model the dynamics by a Lagrangian formulation and apply torque components to dissipate motion. Differential angula.r rates between teleoperator and object are assumed, and control responses after capture are reviewed. Introduction DEVELOP~ICrof the space shuttle system will require an entirely new philosophy toward handling of payloads in orbit. The use of remote manipulators will provide the capability to deploy, retrieve, and repair satellites. Along with this new spectrum of capabilities comes a new set of problems associated with maneuvering masses in orbit. Typically, a satellite to be retrieved can be expected to have some angular motion. Thus, retrieval schemes must be designed to anticipate such situations. Angular momentum must be eliminated before repairs or return can be effected. The process of nulling angular rates is sometimes called "passivation." Payloads which have been identified for potential retrieval operations include: (a) stabilized, normally-operating satellites which require periodic servicing; (b) freely tumbling or spinning payloads without active attitude control; and (c) satellites that have developed an attitude control malfunction and have resulted in some unanticipated state of motion. Typical satellites to be launched in the 1980s and 1990s have been previously identified [1]. Of these, the Research Applications Module (RAM) appears to be one of the most demanding in terms of size and inertia for retrieval missions. Such a payload is of maximum size for the shuttle cargo bay and will require extremely precise handling. It is envisioned as having a control moment gyro attitude system, which could cause random motion if a failure occurs during the mission. Its configuration and coordinate system are depicted in Fig. 1. The dynamic state of a satellite being retrieved could range from a completely stabilized one to a situation of general tumbling. A body is said to be spinning when angular momentum and angular velocity vectors are parallel. This corresponds to spin about the major or minor axis of inertia. To further categorize motion, spin about a symmetry axis may be called "simple spin" and about a transverse axis, "flat spin". A general state of "tumble" exists when angular velocity components about all three body principal axes are of the same order of magnitude[2]. In many cases a body is said to be in "nutational" motion. This is characterized by having the angular velocity vector close to the major or minor axis such that transverse velocity is small. Although general attitude motion of a passive body will always degenerate into spin about its major principal axis if given enough time, there are situations which require retrieval of satellites ?This work has been supported by NASAGrants NGR 39-009-162and NSG-7078.. $Departmentof AerospaceEngineering,The PennsylvaniaState University,UniversityPark, PA 16802,U.S.A. 57

58

cez~ ~

H3

Xl

\

I ~ u m 1. Schematic of research applications module. possessing nutational motion. The problem of docking with such a body is a complicated one. Special end effectors and multiple degrees of freedom are required of the grappling mechanisms. Previous studies have considered requirements for shuttle attached manipulators [3] and free-flying teleoperators[l]. Specific objectives included retrieval performance for typical satellite configurations and dynamic states. One study [4] considered the problem of passivation of a spinning object with nutation assuming a multi-degree of freedom arm. However, none of these investigations considered effects of misaligument during capture. Furthermore, there are some basic stability questions that must be answered about the dynamics and control aspects of capture. Problems related to misaligument, stability, and certain control aspects are identified here. Differential angular rates are assumed during capture and control responses simulated. A Lagrangian formulation is used to develop equations for combined teleoperator-satellite motion, and passivation sequences are demonstrated via computer.

System Dynamics A retrieval sequence begins by deploying the teleoperator from the shuttle bay. When it reaches the vicinity of the satellite, one or two circumnavigations may be necessary for inspection and determination of angular momentum state. This information is essential for alignment during the capture phase. As mentioned above, the RAM spacecraft will be considered for retrieval. Expected yaw, pitch, and roll rates, co,, ~o,, o~,,for various failure modes have been estimated as follows: (i) An abnormal shut-down of the attitude control system (ACS) followed by an extended period of satellite drift would result in ~o, = co, ~<0.025 rad/sec. ¢0, ~ 0.1 rad/sec. (ii) A failure of the AC~ about one axis leads to (a) Failure in roll ~o~,~o, ~<0.1 rad/sec. ~0z ~<10 rad/sec. (b) Failure in pitch or yaw cox,~o, ~ 1 rad/sec. ~o, ~ I rad/sec.

59

(iii) A failure of the satellite ACS about more than one axis would result in ~x, ~, ~ l rad/sec. oJz <~ 10 rad/sec. The nutation cone angle 0 depends upon the ratio of the combined pitch-yaw rate (o~,) to the roll rate (~oz). This coning angle is zero for a pure spin (aJ,, = 0). When the coning angle is small, the preferred approach for capture with convenience may be along the momentum vector to the end face of the satellite (end approach). However, when this angle is large, the preferred approach may be to the waist of the satellite at the center of mass (waist approach). The rendezvous phase ends with the teleoperator about 6--15 m away from the satellite along the final approach direction, as established by the inspection maneuver. Consider the cylindrical satellite of Fig. 1, which is both spinning and tumbling in space. When these angular rates are passive, i.e. there are no external torques acting on the satellite initially, the total angular momentum vector H of the satellite is fixed in inertial space. The body continues to spin about its longitudinal axis and to precess or cone about the angular momentum vector at a constant coning angle 0. In the absence of any external torques and energy dissipation the coning angle 0 and spin and precession rates remain constant for a symmetrical satellite. Passivation can be accomplished by applying torques through teleoperator gripping arms in directions opposite to those of the ~o components. There are obviously many ways of applying torques to the body, depending upon the design and geometry of teleoperator arms. Three ways of nullifying components of momentum may be identified: (1) eliminate either the spin component or the precession component and Olen the other, (2) eliminate both components simultaneously by applying torques on spin and precession axes or (3) eliminate both components simultaneously by applying torques on spin and precession axes and a torque to simultaneously reduce the coning angle. This last procedure was found to passivate at the fastest rate. To eliminate the precession component, ~, of momentum first, apply a torque in the opposite direction about Z. This torque will cause some drift of the momentum vector, which can be minimized by using a small torque magnitude. As ~ reaches zero, the remaining motion should be close to simple spin. During succeeding despin, 0 will increase somewhat. Thus, a succession of decone and despin will be required for complete passivation. If despin is executed first, this procedure is reversed. Figure 2 illustrates a potential grappler configuration with two degrees of freedom in the wrist and two in the shoulder. In method (2), both despinning and detumbling torques are applied simultaneously. If properly executed momentum drift can be eliminated, i.e. the resultant torque must oppose H. In approach (3), the satellite is again simultaneously despun and detumbled, but without having to adjust torque magnitudes to maintain momentum direction. After capture and passivation, the teleoperator returns to the shuttle.

Equations of Motion The equations of motion for the rigid satellite under the influence of despinning and detumbling torques, and for the combined teleoperator-satellite after capture, are now considered. Coordinate systems and nomenclature are illustrated in Figs. 1 and 2. The X, Y, Z system is the orbit frame with X outward along the radius vector, Y normal to the orbit, and Z in the orbit to complete a right-hand system. Satellite fixed coordinates are x, y, z, with z being the longitudinal axis as shown in Fig. 1. These axes are related to X, Y, Z through a set of Euler angles, ~, 0, dp. Figure 2 illustrates the coordinate frames used in the combined teleoperator. Here x, y, z is fixed to the shoulder of the teleoperator. Consider first the satellite under the influence of despinning and detumbling torques, using the Lagrangian formulation, express total kinetic energy of a symmetric tumbling body as 1

T, = ~ [I,,~, ~ + l , , ~ / +

Using the geometry of Fig. 1,

I,,~?].

(1)

60

(2) os~, = os,,2 + oo, e = 62 sine 0 + 02.

(3)

Noting that/~. = I... yields 1 "z 1 O]=+~lx.O +~ I,.[~ +~ cos 0l e.

T, = ~h.[~ sin

(4)

Lagrange's equations are

d-t" ~

-~

= ~" (r=1,2,3)

(5)

where L = T - V, and ~', = generalized force (or torque) corresponding to the rth generalized coordinate. The potential energy, V is zero here. The three generalized coordinates are O, 0 and 0. Substituting expression (4) into (5) yields the equations of satellite motion (lx. s i n e O + I = . c o s e O ) O + 2 ( I z . - l = . ) s i n O c o s O O ~ + I = . ( ~ c o s O - ~ s i n O O ) = r ,

(6)

I..§ + (I.. - Ix.)~ =sin 0 cos 0 + I . . ~ sin 0 = ~'~

(7)

I=.(~; + ¢; cos 0 - ¢J0 sin 0) = r~.

(8)

These second-order, coupled differential equations were solved on a digital computer with various combinations of despinning and detumbling torque functions, including constant magnitudes, torques proportional to angular rates or coning angle, etc. Again using the Lagrangian formulation the equations of motion for the combined teleoperator-satellite were generated and solved via digital computer. It should be noted that the grappler of Fig. 2 is unstable in the configuration shown because of dynamic unbalances due to mass distribution. For example, if its arm and gripping hand are spun to the expected RAM angular rates, then the teleoperator would tumble in a matter of seconds without extremely excessive attitude control functions. This situation can be corrected by adding counter masses

/

Z

/

q;

J

lr 3

~2

•

"r t

~2 ¥ Y

X

Rgum 2. Grappler configuration for capturing small payloads about their waist.

61 which articulate to maintain dynamic balance. Such balance mechanisms were assumed when results were obtained. Initial conditions were set by assuming a misalignment angle between the momentum vectors of teleoperator and satellite. Upon capture of the satellite, the momentum vector shifts due to this misalignment, resulting in an angle 0, between H,, the resultant momentum, and z axis, as shown in Fig. 3. This angle was calculated by assuming the satellite is gripped firmly and it picks up the rotational rates of this grappler, because its mass is much less than that of the teleoperator. Results Results were obtained for both the satellite undergoing various torque combinations and the motion for the combined teleoperator-satellite. Consider first a RAM with properties listed in Table 1. Initial Euler rates and angles were taken as 0 = 0.46tad/see, ~ = 4.0 tad/see, and 0 = 65.9°. Figure 4 shows the motion due to a constant despin (~) torque of magnitude 2700 N-m. Z

Hr 2'

Oi

x

Flgure 3.

Coordinate axes for the motion of combined teleoperator-satellite.

5

05

4

04

3

0.3

X%__~

8

,.. ,.,,." •

.~

sto~ped Ot 2 8 sec

"~

I Q:>

-..+ 0

0

J 0

~0

I

20

I

i

30

40

Figure 4. Motion of RAM due to a constant despin torque. Table 1. Estimated mass and inertia properties of RAM Mass = 9072 k g Length = 18 m Dia.= 3.4m I = = 6616 k g - m =

MMT VOL. 12 NO, I--E

I.

= 6616 k g - m 2

l.z

= 297 k g - m 2

5O

62 The spin rate is reduced to nearly zero in 28 sec. A flat spin follows with a rate equal to the original precession rate. Figure 5 illustrates the effect of a constant deconing torque of the same magnitude. Apparently, the spin rate is not changed and coning continues. This is due to the effects of gyrocompassing. Figure 6 shows the effect of applying this constant torque about the 0 direction. Spin rate is again unchanged, but precession dissipates in 23 sec and 0 goes to 26°. Finally, constant torques were applied about 0 and ~ directions with results shown in Fig. 7. Spin rate is eliminated in 28 sec and ~ goes to 0.1 radlsec at this time. However, 0 goes to 81°. Simulations of the combined teleoperator-satellite were also obtained. Figure 8 shows the motion of 0, for the case of no misalisnments or differential rates during capture. Precession and

5

05

4

04

-

3

u 03

o 2

02

U

-

8 •"e-

~

l

i

q

~

o

2

i(l)

~

._> I

OI

o

o

l

1

5

05

4

O4

'-~

'

I

L

0 I0 20 30 Figure 5. Motion of RAM due to constant deconing torque.

4O

50

I

1:11t

02

01

0

0

I0

0

20

Time,

30

I 40

50

sec

Figure 6. Motion of RAM due to an antiprecession torque.

5

0

5

4

04

3

0.3

~

~

~,.~ q

.-e.

->

0

I

0

0

~

~0

20 Time.

30

] 40

50

sec

Rgure 7. Motion of RAM due to constant precession and despin torques.

63

0.03f 0.02

L

4o ~o o

T i m e , sec -O.OI

-O.C~

-0.03

L

Rgure 8. Motion of combined system with no misalignments.

spin rates of this combined system are constant, but the coning angle 0, is oscillatory in nature. However, the maximum value of 0, is less than 0.55 °. This should be manageable by the attitude control system. Figure 9 depicts the history of 0~ for a differential spin rate of 30% during capture. Precession and spin rates are constant after capture, but 0, undergoes slight oscillations. Other misalignments and differential rates were tried, but no catostrophic events occurred for reasonable expected values of capture errors. Conclusions The results of a detailed study of symmetric satellite motion under the influence of various combinations of passivating torques have been presented. Also, the motion of a combined teleoperator-satellite system has been discussed. Of the many torque functions that could be applied, a constant torque passivated motion in its direction of application faster than other forms. For passivating the satellite motion simultaneously about all axes, i.e. ~, 0 and 4;, it was

003t 0 02

00'

30 I

0

~--0.o~

Time,

sec

-002

-003Flgure 9. Motion of combined system for 30% differential spin rate.

64 found that the best method would be to apply a constant torque on all the three axes, discontinue the despinning torque when the spin rate goes to zero, discontinue the detumbling torque when the precession is nullified, and finally decone with a deconing torque. The whole operation would take about 40 sec for a RAM size spacecraft. It is concluded that no general procedure would suit all satellites. A particular procedure of passivating should be developed after observing the approximate spin and tumble rates of the particular object to be retrieved. The maximum coning angle after the capture, even in the worst case, was well within the capability of typical attitude control systems. References I. G. T. Onega and J. H. Clingman, Free-Flying teleoperator requirements and conceptual design. In Proc. of the First National Conference on Remotely Manned Systems (edited by E. Hecr) 19--32.California Institute of Technology (1973). 2. M. H. Kaplan, The problem of docking with a passive orbiting object which possesses angular momentum. In Astronautical Research 1971, (Edited by L. G. Napolitano) 203-217. Reide|, New York (1973). 3. C. E. Bodey and F. J. Cepollina, Shuttle-attached manipulator system requirements. In Proc. of the First National Conference on Remotely Manned Systems, (Edited by E. Heer) 77-83, California Institute of Technology (1973). 4. G. C. Faile, D. N. Counter and E..1. Bourgeois, Dynamic passivation of a spinning and tumbling satellite using free-flying teleoperators. In Proc. of the First National Conference on Remotely Manned Systems (Edited by E. Heer), pp. 63-73. California Institute of Technology (1973).

8T~a~UN~-

M

STABILITATS~RO~

F~R~.~TEUER~EH

OEBITALER A ~ G U N G

M. H. Kaplan und A. A. Nadkazni -

Die Entwic~lu~6 sines Raumf~h~s~s~ems wird eine neue Philosoph£e

im U~ang mat o~bitalen Hutzlas~en e r f o ~ e ~ n . Die Anwendung ferngesteuer~er Gerite wlzd MSgllchkeiten schaffen, Satelllten zu installieren, aufzubrlngen und zu repa~leren. Zusa~Ben mit diesem neuen F~igkeitsspektrum werden wLr einen K~e£s yon Problemen in Zusam~e~har~ mat dsr ManSv~£erun~ von Massen in Umlaufbahnen antreffen. Typlsshe~eise unterlieg~ eln Satellit, der aufgebracht ws~den soll, einer D=ehbewegung. Dea~alb mt~esen Aufbrln6un6smodelle entworfen .erden, die elne solchs Situation ber~cksichtlgen. Drall mu~ el~-

miniert we~den, bevor ~£e Repa~a~ur oder die R~ckkehr zur ~ d e be~innen kann. Dieser l~o~eB de= Eliminis=ung unerw~nsohtsr Drehbewegungen w~rd manchmal auch "Passivlerun6" ~enannt. ~u~zlasten, dle ftt~ Aufbrln6ungsoperatlonsn

in

Frage ~o"~en, u~fassen! (a) s t a b i l i s i e r t e , n o ~ a l operie=ende S a t e l l i t e n , die pe~iod~schen Service erfo=dern, (b) f r e i taumelnde oder f=ei d~ehende N~tzlasten ohne aktive La~es~euerun6 und (c) Sa~elliten mit fehlerhaft ar-

beitender La6esteuerun~, ~e~ultierend aus einem nicht zu erwartenden Bewe6un~szustand. ~as C ~ e und Massent~6~elt betrlf~t, so erscheint h£er des vor~eschla6ene Raumfa~rtlabozatorium e£ner der he~ausra~enden Satalliten zu se£n. Solch eine Nu~zlast ist yon =axlmaler C ~ e ft~ den Laderaum der F~ire und erfordert ex~re~ genaues Handeln w~hrend der Au~br£n~un6sm£ss~on Es ist beabsichti~t, dee Raumfa~rtlaboratorlum mit e£nem Krelsellagesteue-

~ungssystem a u s z u ~ s ~ e n , das i~n Falle eines wah~end der Mission a u f t r e t e n den Fehlers e£nen nicht ~orhersehba~en Bewe~un&szustand v¢~ursac.~t. Das Problem des Ankoppels an solcn einen K~rper ist ~ompl£zlert. Spezielle Grel£hande und mehrere Freiheits6rade sind fu/ den Grei£mecnanis~us zu £or~ern. Die Er6ebnisse yon Un~ers~chun~en betre~end "~fekte yon l ehl~r~a~ter Ann~heru~6 ~ah~end der Au~bz£n~un~ werden besc~£eoen. .,e£ternin mds~en eAni6e ~undle~ende Stabilit~tsfra6en betref~end DynamiM und Steuerun~saspe~te der Erg~e£fung beantworte~ werden. Problems In Zusa~ennan~ ~£t ~ehler~s~te~ Ann~herun~, Stabil£t:~ und ~ewis~e Steu~run6saspe~te ~erden identifizie~t. Differenzen £n de~ Dre~bewe6un6 w~o/end der Er~e~fun~ ~erden an6eno~en, und die Steuerun6santworten werden slmu!lert. Zttr Eut~icklun~ der Gleichun6en der ~ombinierten ~ewe6un6 z~,ischen fern~esteuertem Gerat und Satellit wird C Le La6~an~esohe ~etrachtun~sme£se benutzt. Der Bewe~ncsablauf whhrend der Pass£vierung wird mit Hilfe eines Computers slmuliert.