DYNAMICS AND CONTROL OF MULTIBODY TETHERED SYSTEMS …

D Y N A M I C S A N D C O N T R O L OF M U L T I B O D Y T E T H E R E D SYSTEMS USING A N O R D E R - N F O R M U L A T I O N

S P I R O S K A L A N T Z I S

B Eng (Honours) McGill University Montreal Canada 1994

A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F

M A S T E R O F A P P L I E D S C I E N C E

in

The Faculty of Graduate Studies Department of Mechanical Engineering

We accept this thesis as conforming to the required standard

T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A

September 1996

copy Spiros Kalantzis 1996

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Department of

The University of British Columbia Vancouver Canada

DE-6 (288)

ABSTRACT

The equations of motion for a multibody tethered satellite system in three dishy

mensional Keplerian orbit are derived The model considers a multi-satellite system

connected in series by flexible tethers Both tethers and subsatellites are free to unshy

dergo three dimensional attitude motion together with deployment and retrieval as

well as longitudinal and transverse vibration for the tether The elastic deformations

of the tethers are discretized using the assumed mode method The tether attachment

points to the subsatellites are kept arbitrary and time varying The model is also cashy

pable of simulating the response of the entire system spinning about an arbitrary

axis as in the case of OEDIPUS-AC which spins about the nominal tether length

or the proposed BICEPS mission where the system cartwheels about the orbit norshy

mal The governing equations of motion are derived using a non-recursive order(N)

Lagrangian procedure which significantly reduces the computational cost associated

with the inversion of the mass matrix an important consideration for multi-satellite

systems Also a symbolic integration and coding package is used to evaluate modal

integrals thus avoiding their costly on-line numerical evaluation

Next versatility of the formulation is illustrated through its application to

two different tethered satellite systems of contemporary interest Finally a thruster

and momentum-wheel based attitude controller is developed using the Feedback Linshy

earization Technique in conjunction with an offset (tether attachment point) control

strategy for the suppression of the tethers vibratory motion using the optimal Linshy

ear Quadratic Gaussian-Loop Transfer Recovery method Both the controllers are

successful in stabilizing the system over a range of mission profiles

ii

T A B L E OF C O N T E N T S

A B S T R A C T bull i i

T A B L E O F C O N T E N T S i i i

L I S T O F S Y M B O L S v i i

L I S T O F F I G U R E S x i

A C K N O W L E D G E M E N T xv

1 I N T R O D U C T I O N 1

11 Prel iminary Remarks 1

12 Brief Review of the Relevant Literature 7

121 Mul t ibody O(N) formulation 7

122 Issues of tether modelling 10

123 Att i tude and vibration control 11

13 Scope of the Investigation 12

2 F O R M U L A T I O N O F T H E P R O B L E M 14

21 Kinematics 14

211 Prel iminary definitions and the itfl position vector 14

212 Tether flexibility discretization 14

213 Rotat ion angles and transformations 17

214 Inertial velocity of the ith link 18

215 Cyl indr ica l orbital coordinates 19

216 Tether deployment and retrieval profile 21

i i i

22 Kinetics and System Energy 22

221 Kinet ic energy 22

222 Simplification for rigid links 23

223 Gravitat ional potential energy 25

224 Strain energy 26

225 Tether energy dissipation 27

23 O(N) Form of the Equations of Mot ion 28

231 Lagrange equations of motion 28

232 Generalized coordinates and position transformation 29

233 Velocity transformations 31

234 Cyl indrical coordinate modification 33

235 Mass matrix inversion 34

236 Specification of the offset position 35

24 Generalized Control Forces 36

241 Preliminary remarks 36

242 Generalized thruster forces 37

243 Generalized momentum gyro torques 39

3 C O M P U T E R I M P L E M E N T A T I O N 42

31 Preliminary Remarks 42

32 Numerical Implementation 43

321 Integration routine 43

322 Program structure 43

33 Verification of the Code 46

iv

331 Energy conservation 4 6

332 Comparison with available data 50

4 D Y N A M I C S I M U L A T I O N 53


42 Parameter and Response Variable Definitions 53

43 Stationkeeping Profile 56

44 Tether Deployment 66

45 Tether Retrieval 71

46 Five-Body Tethered System 71

47 B I C E P S Configuration 80

48 O E D I P U S Spinning Configuration 88

5 A T T I T U D E A N D V I B R A T I O N C O N T R O L 93

51 Att i tude Control 93

511 Prel iminary remarks 93

512 Controller design using Feedback Linearization Technique 94

513 Simulation results 96

52 Control of Tethers Elastic Vibrations 110


522 System linearization and state-space realization 1 1 0

523 Linear Quadratic Gaussian control

with Loop Transfer Recovery 114


v

6 C O N C L U D I N G R E M A R K S 1 2 4

61 Summary of Results 124

62 Recommendations for Future Study 127

B I B L I O G R A P H Y 128

A P P E N D I C E S

I T E N S O R R E P R E S E N T A T I O N O F T H E E Q U A T I O N S O F M O T I O N 133


12 Mathematical Background 133

13 Forcing Function 135

II R E D U C E D E Q U A T I O N S O F M O T I O N 137

II 1 Preliminary Remarks 137

II2 Derivation of the Lagrangian Equations of Mot ion 137

v i

LIST OF SYMBOLS

A j intermediate length over which the deploymentretrieval profile

for the itfl l ink is sinusoidal

A Lagrange multipliers

EUQU longitudinal state and measurement noise covariance matrices

respectively

matrix containing mode shape functions of the t h flexible link

pitch roll and yaw angles of the i 1 link

time-varying modal coordinate for the i 1 flexible link

link strain and stress respectively

damping factor of the t h attitude actuator

set of attitude angles (fjj = ci

structural damping coefficient for the t h l ink EjJEj

true anomaly

Earths gravitational constant GMejBe

density of the ith link

fundamental frequency of the link longitudinal tether vibrashy

tion

A tethers cross-sectional area

AfBfCjDf state-space representation of flexible subsystem

Dj inertial position vector of frame Fj

Dj magnitude of Dj mdash mdash

Df) transformation matrix relating Dj and Ds

D r i inplane radial distance of the first l ink

DSi Drv6i DZ1 T for i = 1 and DX Dy poundgt 2 T for 1

m

Pi

w 0 i

v i i

Dx- transformation matrix relating D j and Ds-

DXi Dyi Dz- Cartesian components of D

DZl out-of-plane position component of first link

E Youngs elastic modulus for the ith l ink

Ej^ contribution from structural damping to the augmented complex

modulus E

F systems conservative force vector

FQ inertial reference frame

Fi t h link body-fixed reference frame

n x n identity matrix

alternate form of inertia matrix for the i 1 l ink

K A Z $i(k + dXi+1)

Kei t h link kinetic energy

M(q t) systems coupled mass matrix

Mma Mma Mmy- control moments in the pitch roll and yaw directions respec-

tively for the t h link

Mr fr rigid mass matrix and force vector respectively

Mred fred- Qred mass matrix force vector and generalized coordinate vector for

the reduced model respectively

Mt block diagonal decoupled mass matrix

symmetric decoupled mass matrix

N total number of links

0N) order-N

Pi(9i) column matrix [T^giTpg^T^gi) mdash

Q nonconservative generalized force vector

Qu actuator coupling matrix or longitudinal L Q R state weighting

matrix

v i i i

Rdm- inertial position of the t h link mass element drrii

RP transformation matrix relating qt and tf

Ru longitudinal L Q R input weighting matrix

RvRn Rd transformation matrices relating qt and q

S(f Mdc) generalized acceleration vector of coupled system q for the full

nonlinear flexible system

th uiirv l u w u u u maniA

th

T j i l ink rotation matrix

TtaTto control thrust in the pitch and roll direction for the im l ink

respectively

rQi maximum deploymentretrieval velocity of the t h l ink

Vei l ink strain energy

Vgi link gravitational potential energy

Rayleigh dissipation function arising from structural damping

in the ith link

di position vector to the frame F from the frame F_i bull

dc desired offset acceleration vector (i 1 l ink offset position)

drrii infinitesimal mass element of the t h l ink

dx^dy^dZi Cartesian components of d along the local vertical local horishy

zontal and orbit normal directions respectively

F-Q mdash

gi r igid and flexible position vectors of drrii fj + ltEraquolt5

ijk unit vectors 1 0 0 r 010-^ and 00 l r respectively

li length of the t h l ink

raj mass of the t h link

nfj total number of flexible modes for the i 1 l ink nXi + nyi + nZi

nq total number of generalized coordinates per link nfj + 7

nqq systems total number of generalized coordinates Nnq

ix

number of flexible modes in the longitudinal inplane and out-

of-plane transverse directions respectively for the i ^ link

number of attitude control actuators

- $ T

set of generalized coordinates for the itfl l ink which accounts for

interactions with adjacent links

qv---QtNT

set of coordinates for the independent t h link (not connected to

adjacent links)

rigid position of dm in the frame Fj

position of centre of mass of the itfl l ink relative to the frame Fj

i + $DJi

desired settling time of the j 1 attitude actuator

actuator force vector for entire system

flexible deformation of the link along the Xj yi and Zj direcshy

tions respectively

control input for flexible subsystem

Cartesian components of fj

actual and estimated state of flexible subsystem respectively

output vector of flexible subsystem

LIST OF FIGURES

1-1 A schematic diagram of the space platform based N-body tethered

satellite system 2

1-2 Associated forces for the dumbbell satellite configuration 3

1- 3 Some applications of the tethered satellite systems (a) multiple

communication satellites at a fixed geostationary location

(b) retrieval maneuver (c) proposed B I C E P S configuration 6

2- 1 Vector components of the i 1 and (i- l ) 1 chain links 15

2-2 Vector components in cylindrical orbital coordinates 20

2-3 Inertial position of subsatellite thruster forces 38

2- 4 Coupled force representation of a momentum-wheel on a rigid body 40

3- 1 Flowchart showing the computer program structure 45

3-2 Kinet ic and potentialenergy transfer for the three-body platform based

tethered satellite system (a) variation of kinetic and potential energy

(b) percent change in total energy of system 48

3-3 Kinet ic and potential energy transfer for the five-body platform based



3-4 Simulation results for the platform based three-body tethered

system originally presented in Ref[43] 51

3- 5 Simulation results for the platform based three-body tethered

system obtained using the present computer program 52

4- 1 Schematic diagram showing the generalized coordinates used to

describe the system dynamics 55

4-2 Stationkeeping dynamics of the three-body STSS configuration

without offset (a) attitude response (b) vibration response 57


xi

with offset along the local vertical

(a) attitude response (b) vibration response 60


with offset along the local horizontal



with offset along the orbit normal



wi th offset along the local horizontal local vertical and orbit normal


4-7 Deployment dynamics of the three-body STSS configuration

without offset





4-9 Retrieval dynamics of the three-body STSS configuration

with offset along the local vertical and orbit normal


4-10 Schematic diagram of the five-body system used in the numerical example 77

4-11 Stationkeeping dynamics of the five-body STSS configuration

without offset


4-12 Deployment dynamics of the five-body STSS configuration

wi th offset along the local vertical


4-13 Stationkeeping dynamics of the three-body B I C E P S configuration

x i i



4-14 Cartwheeling dynamics with deployment for the three-body B I C E P S



4-15 Spin dynamics (7 = l deg s ) of the three-body O E D I P U S configuration



4- 16 Spin dynamics (7 = 10deg s ) of the three-body O E D I P U S configuration



5- 1 Controlled dynamics of the three-body STSS during stationkeeping

using the nonlinear rigid F L T controller with offset along the

local horizontal local vertical and orbit normal

(a) attitude and vibration response (b) control actuator time histories 98

5-2 Controlled dynamics of the three-body STSS during stationkeeping

using the nonlinear flexible F L T controller with offset along the




using the nonlinear rigid F L T controller with offset along

the local vertical


5-4 Deployment dynamics of the three-body STSS using the nonlinear

rigid F L T controller with offset along the local vertical


5-5 Retrieval dynamics of the three-body STSS using the non-linear

rigid F L T controller with offset along the local vertical and orbit normal


x i i i

5-6 Block diagram for the L Q G L T R estimator based compensator 115

5-7 Singular values for the L Q G and L Q G L T R compensator compared to target

return ratio (a) longitudinal design (b) transverse design 118


using the nonlinear rigid F L T attitude controller and L Q G L T R

offset vibration controller (a) attitude and libration controller response

(b) vibration and offset response 122

xiv

A C K N O W L E D G E M E N T

Firs t and foremost I would like to express the most genuine gratitude to

my supervisor Prof V i n o d J M o d i whose unending guidance and encouragement

proved most invaluable during the course of my studies

I am also deeply indebted to Dr Satyabrata Pradhan for his limitless patience

and technical advice from which this work would not have been possible I would also

like to extend a word of appreciation towards Prof A r u n K Mis ra ( M c G i l l University)

for ini t iat ing my interest in space dynamics and control

Also I would like to thank my collegues Dr Sandeep Munshi M r Mathieu

Caron and M r Yuan Chen as well as Dr Mae Seto M r Gary L i m and M r Mark

Chu for their words of advice and helpful suggestions that heavily contributed to all

aspects of my studies

Final ly I would like to thank all my family and friends here in Vancouver and

back home in Montreal whose unending support made the past two years the most

memorable of my life

The research project was supported by the Natural Sciences and Engineering

Research Council s P G S - A Scholarship held by the author as well as by N S E R C grant

A-2181 held by Prof V J M o d i

xv

1 INTRODUCTION

11 Prel iminary Remarks

W i t h the ever changing demands of the worlds population one often wonders

about the commitment to the space program On the other hand the importance

of worldwide communications global environmental monitoring as well as long-term

habitation in space have demanded more ambitious and versatile satellites systems

It is doubtful that the Russian scientist Tsiolkovsky[l] when he first proposed the

uti l izat ion of the Earths gravity-gradient environment in the last century would have

envisioned the current and proposed applications of tethered satellite systems

A tethered satellite system consists of two or more subsatellites connected

to each other by long thin cables or tethers which are in tension (Figure 1-1) The

subsatellite which can also include the shuttle and space station are in orbit together

The first proposed use of tethers in space was associated with the rescue of stranded

astronauts by throwing a buoy on a tether and reeling it to the rescue vehicle

Prel iminary studies of such systems led to the discovery of the inherent instability

during the tether retrieval[2] Nevertheless the bir th of tethered systems occured

during the Gemini X I and X I I missions[3] in 1966 when a short tether (10-30m)

was used to generate the first artificial gravity environment in space (000015g) by

cartwheeling (spinning) the system about its center of mass

The mission also demonstrated an important use of tethers for gravity gradient

stabilization The force analysis of a simplified model illustrates this point Consider

the dumbbell satellite configuration orbiting about the centre of Ear th as shown in

Figure 1-2 Satellite 1 and satellite 2 are connected to each other by a tether with

1

Space Station (Satellite 1 )

2

Satellite N

Figure 1-1 A schematic diagram of the space platform based N-body tethered satellite system

2

Figure 1-2 Associated forces for the dumbbell satellite configuration

3

the whole system free to librate about the systems centre of mass C M The two

major forces acting on each body are due to the centrifugal and gravitational effects

Since body 1 is further away from the earth its gravitational force Fg^y is smaller

than that of body 2 ie Fgi lt Fg^ However its centrifugal force FCl is greater

then the centrifugal force experienced by the second satellite (FCl gt FC2) Moreover

for body 1 Fg^ lt Fc^ in contrast to Fg2 gt FC2 for body 2 Thus resolving the force

components along the tether there is an evident resultant tension force Ff present

in the tether Similarly adding the normal components of the two forces vectorially

results in a force FR which restores the system to its stable equil ibrium along the

local vertical These tension and gravity-gradient restoring forces are the two most

important features of tethered systems

Several milestone missions have flown in the recent past The U S A J a p a n

project T P E (Tethered Payload Experiment) [4] was one of the first to be launched

using a sounding rocket to conduct environmental studies The results of the T P E

provided support for the N A S A I t a l y shuttle based tethered satellite system referred

to as the TSS-1 mission[5] This experiment aimed to study the electrodynamic effects

of a shuttle borne satellite system with a conductive tether connection where an

electric current was induced in the tether as it swept through the Earth s magnetic

field Unfortunetely the two attempts in August 1992 and February 1996 resulted

in only partial success The former suffered from a spool failure resulting in a tether

deployment of only 256m of a planned 20km During the latter attempt a break

in the tethers insulation resulted in arcing leading to tether rupture Nevertheless

the information gathered by these attempts st i l l provided the engineers and scientists

invaluable information on the dynamic behaviour of this complex system

Canada also paved the way with novel designs of tethered system aimed

primari ly at the study of the ionosphere and Northern Lights (Aurora Borealis)

4

The O E D I P U S A and C (Observation of Electrified Distributions in the Ionospheric

P lasma - a Unique Strategy) missions[6] launched from a sounding rocket in January

1989 and November 1995 respectively provided insight into the complex dynamical

behaviour of two comparable mass satellites connected by a 1 km long spinning

tether

Final ly two experiments were conducted by N A S A called S E D S I and II

(Small Expendable Deployable System) [7] which hold the current record of 20 km

long tether In each of these missions a small probe (26 kg) was deployed from the

second stage of a D E L T A II rocket thus succesfully demonstrating the feasibility of

long tether deployment Retrieval of the tether which is significantly more difficult

has not yet been achieved

Several proposed applications are also currently under study These include the

study of earths upper atmosphere using probes lowered from the shuttle in a low earth

orbit ( L E O ) to the deployment and retrieval of satellites for servicing Even more

promising application concerns the generation of power for the proposed International

Space Station using conductive tethers The Canadian Space Agency (CSA) has also

proposed a dual-mass satellite system B I C E P S (Bl-static Canadian Experiment on

Plasmas in Space) to make simultaneous measurements at different locations in the

environment for correlation studies[8] A unique feature of the B I C E P S mission is

the deployment of the tether aided by the angular momentum of the system which

is ini t ia l ly cartwheeling about its orbit normal (Figure 1-3)

In addition to the three-body configuration multi-tethered systems have been

proposed for monitoring Earths environment in the global project monitored by

N A S A entitled Mission to Planet Ear th ( M T P E ) The proposed mission aims to

study pollution control through the understanding of the dynamic interactions be-

5

Retrieval of Satellite for Servicing

lt mdash

lt Satellite

Figure 1-3 Some applications of the tethered satellite systems (a) multiple comshymunication satellites at a fixed geostationary location (b) retrieval maneuver (c) proposed BICEPS configuration

6

tween the Earths atmosphere biosphere hydrosphere and cryosphere This wi l l be

accomplished with multiple tethered instrumentation payloads simultaneously soundshy

ing at different altitudes for spatial correlations Such mult ibody systems are considshy

ered to be the next stage in the evolution of tethered satellites Micro-gravity payload

modules suspended from the proposed space station for long-term experiments as well

as communications satellites with increased line-of-sight capability represent just two

of the numerous possible applications under consideration

12 Brief Review of the Relevant Literature

The design of multi-payload systems wi l l require extensive dynamic analysis

and parametric study before the final mission configuration can be chosen This

wi l l include a review of the fundamental dynamics of the simple two-body tethered

systems and how its dynamics can be extended to those of multi-body tethered system

in a chain or more generally a tree topology

121 Multibody 0(N) formulation

Many studies of two-body systems have been reported One of the earliest

contribution is by Rupp[9] who was the first to study planar dynamics of the simshy

plified Shuttle based Tethered Satellite System (STSS) His pioneering work led to

the discovery of pitch oscillation growth during the retrieval phase Later Baker[10]

advanced the investigation to the third dimension in addition to adding atmospheric

effects to the system A more complete dynamical model was later developed and

analyzed by M o d i and Misra[ l l 12] which included deployment and tether flexibility

A comprehensive survey of important developments and milestones in the area have

been compiled and reviewed by Mis ra and Modi[13] The major conclusions based on

the literature may be summarized as follows

bull the stationkeeping phase is stable

7

bull deployment can be unstable if the rate exceeds a crit ical speed

bull retrieval of the tether is inherently unstable

bull transverse vibrations can grow due to the Coriolis force induced during deshy

ployment and retrieval

Misra Amier and Modi[14] were one of the first to extend these results to the

three-satellite double pendulum case This simple model which includes a variable

length tether was sufficient to uncover the multiple equilibrium configurations of

the system Later the work was extended to include out-of-plane motion and a

preliminary reel-rate control law to regulate the librational motion[15] Kesmir i and

Misra[16] developed a general formulation for N-body tethered systems based on the

Lagrangian principle where three-dimensional motion and flexibility are accounted for

However from their work it is clear that as the number of payload or bodies increases

the computational cost to solve the forward dynamics also increases dramatically

Tradit ional methods of inverting the mass matrix M in the equation

ML+F = Q

to solve for the acceleration vector proved to be computationally costly being of the

order for practical simulation algorithms O(N^) refers to a mult ibody formulashy

tion algorithm whose computational cost is proportional to the cube of the number

of bodies N used It is therefore clear that more efficient solution strategies have to

be considered

Many improved algorithms have been proposed to reduce the number of comshy

putational steps required for solving for the acceleration vector Rosenthal[17] proshy

posed a recursive formulation based on the triangularization of the equations of moshy

tion to obtain an 0(N2) formulation He also demonstrates the use of a symbolic

manipulation program to reduce the final number of computations A recursive L a -

8

grangian formulation proposed by Book[18] also points out the relative inefficiency of

conventional formulations and proceeds to derive an algorithm which is also 0(N2)

A recursive formulation is one where the equations of motion are calculated in order

from one end of the system to the other It usually involves one or more passes along

the links to calculate the acceleration of each link (forward pass) and the constraint

forces (backward or inverse pass) Non-recursive algorithms express the equations of

motion for each link independent of the other Although the coupling terms st i l l need

to be included the algorithm is in general more amenable to parallel computing

A st i l l more efficient dynamics formulation is an O(N) algorithm Here the

computational effort is directly proportional to the number of bodies or links used

Several types of such algorithms have been developed over the years The one based

on the Newton-Euler approach has recursive equations and is used extensively in the

field of multi-joint robotics[1920] It should be emphasized that efficient computation

of the forward and inverse dynamics is imperative if any on-line (real-time) control

of the joint is to be achieved Hollerbach[21] proposed a recursive O(N) formulation

based on Lagranges equations of motion However his derivation was primari ly foshy

cused on the inverse dynamics and did not improve the computational efficiency of

the forward dynamics (acceleration vector) Other recursive algorithms include methshy

ods based on the principle of vir tual work[22] and Kanes equations of motion[23]

Keat[24] proposed a method based on a velocity transformation that eliminated the

appearance of constraint forces and was recursive in nature O n the other hand

K u r d i l a Menon and Sunkel[25] proposed a non-recursive algorithm based on the

Range-Space method which employs element-by-element approach used in modern

finite-element procedures Authors also demonstrated the potential for parallel comshy

putation of their non-recursive formulation The introduction of a Spatial Operator

Factorization[262728] which utilizes an analogy between mult ibody robot dynamics

and linear filtering and smoothing theory to efficiently invert the systems mass ma-

9

t r ix is another approach to a recursive algorithm Their results were further extended

to include the case where joints follow user-specified profiles[29] More recently Prad-

han et al [30] proposed a non-recursive Lagrangian algorithm for factorizing the mass

matrix leading to an O(N) formulation of the forward dynamics of the system A n

efficient O(N) algorithm where the number of bodies N in the system varies on-line

has been developed by Banerjee[31]

122 Issues of tether modelling

The importance of including tether flexibility has been demonstrated by several

researchers in the past However there are several methods available for modelling

tether flexibility Each of these methods has relative advantages and limitations

wi th regards to the computational efficiency A continuum model where flexible

motion is discretized using the assumed mode method has been proposed[3233] and

succesfully implemented In the majority of cases even one flexible mode is sufficient

to capture the dynamical behaviour of the tether[34] K i m and Vadali[35] proposed a

so-called bead model or lumped-mass approach that discretizes the tether using point

masses along its length However in order to accurately portray the motion of the

tether a significantly high number of beads are needed thus increasing the number of

computation steps Consequently this has led to the development of a hybrid between

the last two approaches[16] ie interpolation between the beads using a continuum

approach thus requiring less number of beads Finally the tether flexibility can also

be modelled using a finite-element approach[36] which is more suitable for transient

response analysis In the present study the continuum approach is adopted due to

its simplicity and proven reliability in accurately conveying the vibratory response

In the past the modal integrals arising from the discretization process have

been evaluted numerically In general this leads to unnecessary effort by the comshy

puter Today these integrals can be evaluated analytically using a symbolic in-

10

tegration package eg Maple V Subsequently they can be coded in F O R T R A N

directly by the package which could also result in a significant reduction in debugging

time More importantly there is considerable improvement in the computational effishy

ciency [16] especially during deployment and retrieval where the modal integral must

be evaluated at each time-step

123 Attitude and vibration control

In view of the conclusions arrived at by some of the researchers mentioned

above it is clear that an appropriate control strategy is needed to regulate the dyshy

namics of the system ie attitude and tether vibration First the attitude motion of

the entire system must be controlled This of course would be essential in the case of

a satellite system intended for scientific experiments such as the micro-gravity facilishy

ties aboard the proposed Internation Space Station Vibra t ion of the flexible members

wi l l have to be checked if they affect the integrity of the on-board instrumentation

Thruster as well as momentum-wheel approach[34] have been considered to

regulate the rigid-body motion of the end-satellite as well as the swinging motion

of the tether The procedure is particularly attractive due to its effectiveness over a

wide range of tether lengths as well as ease of implementation Other methods inshy

clude tension and length rate control which regulate the tethers tension and nominal

unstretched length respectively[915] It is usually implemented at the deployment

spool of the tether More recently an offset strategy involving time dependent moshy

tion of the tether attachment point to the platform has been proposed[3738] It

overcomes the problem of plume impingement created by the thruster control and

the ineffectiveness of tension control at shorter lengths However the effectiveness of

such a controller can become limited with an exceedingly long tether due to a pracshy

tical l imi t on the permissible offset motion In response to these two control issues a

hybrid thrusteroffset scheme has been proposed to combine the best features of the

11

two methods[39]

In addition to attitude control these schemes can be used to attenuate the

flexible response of the tether Tension and length rate control[12] as well as thruster

based algorithms[3340] have been proposed to this end M o d i et al [41] have sucshy

cessfully demonstrated the effectiveness of an offset strategy to damp undesired v i shy

brations Passive energy dissipative devices eg viscous dampers are also another

viable solution to the problem

The development of various control laws to implement the above mentioned

strategies has also recieved much attention A n eigenstructure assignment in conducshy

tion wi th an offset controller for vibration attenuation and momemtum wheels for

platform libration control has been developed[42] in addition to the controller design

from a graph theoretic approach[41] Also non-linear feedback methods such as the

Feedback Linearization Technique ( F L T ) that are more suitable for controlling highly

nonlinear non-autonomous coupled systems have also been considered[43]

It is important to point out that several linear controllers including the classhy

sic state feedback Linear Quadratic Regulator ( L Q R ) have received considerable

attention[3944] Moreover robust methods such as the Linear Quadratic Guas-

s ian Loop Transfer Recovery ( L Q G L T R ) method have also been developed and imshy

plemented on tethered systems[43] A more complete review of these algorithms as

well as others applied to tethered system has been presented by Pradhan[43]

13 Scope of the Investigation

The objective of the thesis is to develop and implement a versatile as well as

computationally efficient formulation algorithm applicable to a large class of tethered

satellite systems The distinctive features of the model can be summarized as follows

12

bull TV-body 0(N) tethered satellite system in a chain-type topology

bull the system is free to negotiate a general Keplerian orbit and permitted to

undergo three dimensional inplane and out-of-plane l ibrtational motion

bull r igid bodies constituting the system are free to execute general rotational

motion

bull three dimensional flexibility present in the tether which is discretized using

the continuum assumed-mode method with an arbitrary number of flexible

modes

bull energy dissipation through structural damping is included

bull capability to model various mission configurations and maneuvers including

the tether spin about an arbitrary axis

bull user-defined time dependent deployment and retrieval profiles for the tether

as well as the tether attachment point (offset)

bull attitude control algorithm for the tether and rigid bodies using thrusters and

momentum-wheels based on the Feedback Linearization Technique

bull the algorithm for suppression of tether vibration using an active offset (tether

attachment point) control strategy based on the optimal linear control law

( L Q G L T - R )

To begin with in Chapter 2 kinematics and kinetics of the system are derived

using the O(N) formulation methodology developed by Pradhan et al [30] Chapshy

ter 3 discusses issues related to the development of the simulation program and its

validation This is followed by a detailed parametric study of several mission proshy

files simulated as particular cases of the versatile formulation in Chapter 4 Next

the attitude and vibration controllers are designed and their effectiveness assessed in

Chapter 5 The thesis ends with concluding remarks and suggestions for future work

13

2 F O R M U L A T I O N OF T H E P R O B L E M

21 Kinematics

211 Preliminary definitions and the i ^ position vector

The derivation of the equations of motion begins with the definition of the

inertial position of an arbitrary link of the mult ibody system Let the i ^ link of the

system be free to translate and rotate in 3-D space From Figure 2-1 the position

vector Rdm^ to the mass element drrn on the i ^ link wi th reference to the inertial

frame FQ can be written as

Here D represents the inertial position of the t h body fixed frame Fj relative to FQ

ri = [xiyiZi]T is the rigid position vector of drrii wi th respect to F J and f(fi) is

the flexible deformation at fj also with respect to the frame Fj Bo th these vectors

are relative to Fj Note in the present model tethers (i even) are considered flexible

and X corresponds to the nominal position along the unstretched tether while a and

ZJ are by definition equal to zero On the other hand the rigid bodies referred to as

satellites (i odd) including the platform are taken to be rigid ie f(fj) = 0 T j is

the rotation matrix used to express body-fixed vectors with reference to the inertial

frame

212 Tether flexibility discretization

The tether flexibility is discretized with an arbitrary but finite number of

Di + Ti(i + f(i)) (21)

14

15

modes in each direction using the assumed-mode method as

Ui nvi

wi I I i = 1

nzi bull

(22)

where nXi nyi and n 2 ^ are the number of modes used in the X m and Z directions

respectively For the ] t h longitudinal mode the admissible mode shape function is

taken as 2 j - l

Hi(xiii)=[j) (23)

where l is the i ^ tether length[32] In the case of inplane and out-of-plane transverse

deflections the admissible functions are

ampyi(xili) = ampZixuk) = v ^ s i n JKXj

k (24)

where y2 is added as a normalizing factor In this case both the longitudinal and

transverse mode shapes satisfy the geometric boundary conditions for a simple string

in axial and transverse vibration

Eq(22) is recast into a compact matrix form as

ff(i) = $i(xili)5i(t) (25)

where $i(x l) is the matrix of the tether mode shapes defined as

0 0

î(xiJi) =

^ X bull bull bull ^XA

0

0 0

0

G 5R 3 x n f

(26)

and nfj = nx- + nyi + nZi Note $J(XJJ) is a function of the spatial coordinate x

and not of yi or ZJ However it is also a function of the tether length parameter li

16

which is time-varying during deployment and retrieval For this reason care must be

exercised when differentiating $J(XJJ) with respect to time such that

d d d k) = Jtregi(Xi li) = ^ 7 $ t ( ^ i k)Xi + gf$i(xigt li)k- (2-7)

Since x represents the nominal rate of change of the position of the elemental mass

drrii it is equal to j (deploymentretrieval rate) Let t ing

( 3 d

dx~ + dT)^Xili^ ( 2 8 )

one arrives at

$i(xili) = $Di(xili)ii (29)

The time varying generalized coordinate vector 5 in Eq(25) is composed of

the longitudinal and transverse components ie

ltM) G s f t n f i x l (210)

where 5X^ 5y- and Sz^ are nx- x 1 ny x 1 and nz- x 1 size vectors and correspond to

$ X and $ Z respectively

213 Rotation angles and transformations

The matrix T j in Eq(21) represents a rotation transformation from the body

fixed frame Fj to the inertial frame FQ ie FQ = T j F j It is defined by the 3-2-1

ordered sequence of elementary rotations[46]

1 P i tch F[ = Cf ( a j )F 0

2 R o l l F = C f (3j)F = C7f (3j)Cf ( a j )F 0

3 Yaw F = C7( 7 i )F = C7(7l)cf^)Cf(ai)FQ = Q F 0

17

where

c f (A) =

-Sai Cai

0 0

and

C(7i ) =

0 -s0i] = 0 1 0

0 cpi

i 0 0 = 0

0

(211)

(212)

(213)

Here Eq(211) corresponds to a rotation of ctj (pitch) about the inertial axis ZQ (orbit

normal) of frame FQ resulting in a new intermediate frame F It is then followed by

a roll rotation fa about the axis y of F- giving a second intermediate frame F-

Final ly a spin rotation 7J about the axis z of F- is given resulting in the body

fixed frame Fj for the i lt l link Since all the three rotation matrices are orthogonal

it follows that FQ = C~1Fi = Cj F and hence

C CPi Cai ~ CH Sai + SrYi SPi Sai 5 7 i Sampi + Cli SPi Cci CPi Sai C1i Cai + 5 7 i SPi ~ S1 Ca + C 7 bull Sp Sai

(214) SliCPi CliCPi

Here Cx and Sx are abbreviations for cos(x) and sin(x) respectively

214 Inertial velocity of the ith link

mdash Lett ing pj = fj - f $j(5j and differentiating Eq(21) with respect to time gives

m- D j + T j f j + T^Si + T j pound + T j $ j 5 j (215)

Each of the terms appearing in Eq(215) must be addressed individually Since

fj represents the rigid body motion within the link it only has meaning for the

deployable tether and since yi = Zj = 0 for al l tether links (i even) fj = Ifi where 1

is the unit vector 1 0 0T For the case of rigid links (i odd) fj = 0

18

Evaluating the derivative of each angle in the rotation matrix gives

Tî9i = Taiai9i + Tppigi + T^fagi (216)

where Tx = J j I V Collecting the scaler angular velocity terms and defining the set

Vi = deg^gt A) 7i^ Eq(216) can be rewritten as

T i f t = Pi9i)fji (217)

where Pi(gi) is a column matrix defined by

Pi(9i)Vi = [Taigi^pîT^gilffi (218)

Inserting Eq(29) and Eq(217) into Eq(215) and defining Sj = 2 + leads to

215 Cylindrical orbital coordinates

The Di term in Eq(219) describes the orbital motion of the t h l ink and

is composed of the three Cartesian coordinates DXi A ^ DZi However over the

cycle of one orbit DXi and Dyi vary dramatically by around the order of Earths

radius This must be avoided since large variations in the coordinates can cause

severe truncation errors during their numerical integration For this reason it is

more convenient to express the Cartesian components in terms of more stationary

variables This is readily accomplished using cylindrical coordinates

However it is only necessary to transform the first l inks orbital components

ie Lgti since only D is a generalized coordinates From Figure 2-2 it is apparent

(219)

that

(220)

19

Figure 2-2 Vector components in cylindrical orbital coordinates

20

Eq(220) can be rewritten in matrix form as

cos(0i) 0 0 s i n ( 0 i ) 0 0

0 0 1

Dri

h (221)

= DTlDSv

where Dri is the inplane radial distance 9 the true anomaly and DZl the out-of-

plane distance normal to the orbital plane The total derivative of D wi th respect

to time gives in column matrix form

1 1 1 1 J (222)

= D D l D 3 v

where [cos(î) -Dnsm(8i) 0 ]

(223)

For all remaining links ie i = 2 N

cos(0i) - A - j S i n ^ i ) 0 sin(î) D r i c o s ( 0 i ) 0

0 0 1

DTl = D D i = I d 3 i = 2N (224)

where is the n x n identity matrix and

2N (225)

216 Tether deployment and retrieval profile

The deployment of the tether is a critical maneuver for this class of systems

Cr i t i ca l deployment rates would cause the system to librate out of control Normally

a smooth sinusoidal profile is chosen (S-curve) However one would also like a long

tether extending to ten twenty or even a hundred kilometers in a reasonable time

say 3-4 orbits This is usually difficult or impossible to accomplish solely wi th one

S-curve profile Often a composite S-curve-Steady-S-curve profile is adopted Thus

21

the deployment scheme for the t h tether can be summarized as

k = Tr l - cos (^r(ti - to-)) 1 ltUlt h 2 r V A V

k = Vov tiilttiltt2i

k = IT j 1 - C deg S lkitl ^ ^ lt i lt ^

(226)

where to- tt are the ini t ia l and final times of the deployment or retrieval maneuver

respectively A t j = t - tQ mdash tt mdash t2- and the steady deployment velocity VQ is

calculated based on the continuity of l and l at the specified intermediate times t

and to For the case of

and

Ai = i(ti) - ^(t 0 -) = - ^ ( t 2 i ) (227)

2Ali + li(tf) -h(tQ)

V = lUli l^lL (228)

1 tf-tQ v y

A t = (229)

22 Kinetics and System Energy

221 Kinetic energy

The kinetic energy of the ith link is given by

Kei = lJm RdrnfiÂmi (230)

Setting Eq(219) in a matrix-vector form gives the relation

4 ^ ^ PM) T i $ 4 TiSi]i[t (231)

22

where

m

k J

x l (232)

and nq = -nfj + 7 is the number of generalized coordinates in each link Inserting

Eq(231) into Eq(230) and integrating the kinetic energy for the t h l ink can be

rewritten as

1 T

(233)

where Mti is the links symmetric mass matrix

Mt =

m i D D T D D DDTPifgidmi) DDTTi J ^dm D n T Sidm rn

sym fP(9i)Pi(9i)dmi J P[g^Tîdm J P ( pound ) T ^ m sym sym J ^fîdm f ^f^dmi sym sym sym J sfsidmi

pound Sfttiqxnq

(234)

and mi is the mass of the Ith link The kinetic energy for the entire system ie N

bodies can now be stated as

1 bull T bull 1 bull T Ke = J2 MtA = o laquo Mtqt 2 ^ -i

i=l (235)

where qt = qj^^ bull bull -qfNT a n d Mt is a block diagonal decoupled mass matrix

of the system with Mf- on its diagonal

(236) Mt =

0 0

0 Mt2

0

0 0

MH

0 0 0

0 0 0 bull MtN

222 Simplification for rigid links

The mass matrix given by Eq(234) simplifies considerably for the case of rigid

23

links and is given as

Mti

m D D l

T D D l DD^Piiffidmi) 0

sym 0

0

fP(fi)Pi(i)dmi 0 0 0

0 Id nfj

0 m~

i = 1 3 5 N

(237)

Moreover when mult iplying the matrices in the (11) block term of Eq(237) one

gets

D D l

1 D D l = 1 0 0 0 D n

2 0 (238) 0 0 1

which is a diagonal matrix However more importantly i t is independent of 6 and

hence is always non-singular ensuring that is always invertible

In the (12) block term of M ^ it is apparent that f fidmi is simply the

definition of the centre of mass of the t h rigid link and hence given by

To drrii mdash miff (239)

where fcmi is the position vector of the centre of mass of link i For the case of the

(22) block term a little more attention is required Expanding the integrand using

Eq(218) where ltj = fj gives

fj Ta Tafi fj T Q Tpfi f- T Q T 7 i f j

sym sym

Now since each of the terms of Eq(240) is a scaler or a 1 x 1 matrix it is therefore

equal to its trace ie sum of its diagonal elements thus

tr(ffTai

TTaifi) tr(ffTai

TTpfi) tr(ff Ta^T^fi)

fjT^Tpn rfTp^fi (240)

P(rj)Pj(fi) = syrri trifjTpTTpfi) t r ^ T ^ T ^

sym sym tr(ff T 7 T r 7 f j) J

(241)

Using two properties of the trace function[47] namely

tr(ABC) = tr(BCA) = trCAB) (242)

24

and

j tr(A) =tr(^J A^j (243)

where A B and C are real matrices it becomes quite clear that the (22) block term

of Eq(237) simplifies to

P(fi)Pii)dmi =

tr(Tai

TTaiImi) tTTai

TTpImi) t r (T Q r T 7 7 m ) trTaTTpImi) tr(Tp

TT7Jmi) sym sym sym

(244)

where Im- is an alternate form of the inertia tensor of the t h link which when

expanded is

Imlaquo = I nfi dm bull i sym

sym sym

J x^dmi J Xydmi XZdm sym J yi2dmi f yiZidm

sym J z^dmi J

xVi -L VH sym ijxxi + lyy IZZJ)2

(245)

Note this is in terms of moments of inertia and products of inertia of the t f l rigid

link A similar simplification can be made for the flexible case where T is replaced

with Qi resulting in a new expression for Im

223 Gravitational potential energy

The gravitational potential energy of the ith link due to the central force law

can be written as

V9i f dm f

-A mdash = -A bullgttradelti RdmA

dm

inn Lgti + T j ^ | (246)

where z = 3986 x 105 km 3s 2 (Earths gravitational constant) Expanding binomially

and retaining up to third order terms gives

Vgi ~ ~ D T 2D [J gfgidrrii + 23f T j gidm mdashDT

D2 i l I I Df Tt I grffdmiTiDi

(2-lt 7)

25

where D = D is the magnitude of the inertial position vector to the frame F The

integrals in the above expression are functions of the flexible mode shapes and can be

evaluated through symbolic manipulation using an existing package such as Maple

V In turn Maple V can translate the evaluated integrals into F O R T R A N and store

the code in a file Furthermore for the case of rigid bodies it can be shown quite

readily that

j gjgidrrii = j rjfidrrn = [IXXi + Iyy + IZZi)2 (248)

Similarly the other integrals in Eq(234) can be integrated symbolically

224 Strain energy

When deriving the elastic strain energy of a simple string in tension the

assumptions of high tension and low amplitude of transverse vibrations are generally

valid Consequently the first order approximation of the strain-displacement relation

is generally acceptable However for orbiting tethers in a weak gravitational-gradient

field neglecting higher order terms can result in poorly modelled tether dynamics

Geometric nonlinearities are responsible for the foreshortening effects present in the

tether These cannot be neglected because they account for the heaving motion along

the longitudinal axis due to lateral vibrations The magnitude of the longitudinal

oscillations diminish as the tether becomes shorter[35]

W i t h this as background the tether strain which is derived from the theory

of elastic vibrations[3248] is given by

(249)

where Uj V and W are the flexible deformations along the Xj and Z direction

respectively and are obtained from Eq(25) The square bracketed term in Eq(249)

represents the geometric nonlinearity and is accounted for in the analysis

26

The total strain energy of a flexible link is given by

Ve = I f CiEidVl (250)

Substituting the stress-strain relationship 0 = E(poundi the strain energy is now given

by

Vei=l-EiAi J l l d x h (251)

where E^ is the tethers Youngs modulus and A is the cross-sectional area The

tether is assumed to be uniform thus EA which is the flexural stiffness of the

tether is assumed to be constant Eq(251) is evaluated symbolically for arbitrary

number of modes

225 Tether energy dissipation

The evaluation of the energy dissipation due to tether deformations remains

a problem not well understood even today Here this complex phenomenon is repshy

resented through a simplified structural damping model[32] In addition the system

exhibits hysteresis which also must be considered This is accomplished using an

augmented complex Youngs modulus

EXÊi+jEi (252)

where Ej^ is the contribution from the structural damping and j = The

augmented stress-strain relation is now given as

a = Epoundi = Ei(l+jrldi)ei (253)

where 77^ = EjE^ ltC 1 is defined as the structural damping coefficient determined

experimentally [49]

27

If poundi is a harmonic function with frequency UJQ- then

jet = (254) wo-

Substituting into Eq(253) and rearranging the terms gives

e + I mdash I poundi = o-i + ad (255)

where ad and a are the stress with and without structural damping respectively

Now the Rayleigh dissipation function[50] for the ith tether can be expressed as

(ii) Jo

_ 1 EjAiVdi rU (2-56^ 2 w 0 Jo

The strain rate ii is the time derivative of Eq(249) The generalized external force

due to damping can now be written as Qd = Q^ bull bull bull QdNT where

dWd

^ = - i p = 2 4 ( 2 5 7 )

= 0 i = 13 N

23 0(N) Form of the Equations of Mot ion

231 Lagrange equations of motion

W i t h the kinetic energy expression defined and the potential energy of the

whole system given by Pe = ]Cn=l( 5i+ ej) the equations of motion can be obtained

quite readily using the Lagrangian principle

where q is the set of generalized coordinates to be defined in a later section

Substituting Eqs(235247251 and 257) into Eq(258) leads to the familiar

matr ix form of the non-linear non-autonomous coupled equations of motion for the

28

system

Mq t)t+ F(q q t) = Q(q q t) (259)

where M(q t) is the nonlinear symmetric mass matrix and F(q q t) is the forcing

function which can be written in matrix form as

Q(ltfgt Qi 0 is the vector of non-conservative generalized forces including control inshy

puts acting on the system A detailed expansion of Eq(260) in tensor notation is

developed in Appendix I

Numerical solution of Eq(259) in terms of q would require inversion of the

mass matrix However M is a full matrix of size nqq x nqq where nqq = Nnq is the

total number of generalized coordinates Therefore direct inversion of M would lead

to a large number of computation steps of the order nqq^ or higher The objective

here is to develop an order-N O(N) algorithm for obtaining M _ 1 that minimizes

the computational effort This is accomplished through the following transformations

ini t ia l ly developed for the stationkeeping case by Pradhan et al [30]

232 Generalized coordinates and position transformation

During the derivation of the energy expressions for the system the focus was

on the decoupled system ie each link was considered independent of the others

Thus each links energy expression is also uncoupled However the constraints forces

between two links must be incorporated in the final equations of motion

Let

e Unqxl (261) m

be the set of coupled coordinates of the t h l ink such that q = qT q2 bull bull bull 0T i s the

29

systems generalized coordinates Let qt- be the set of auxiliary decoupled coordinates

defined by Eq(232) such that qt = qj^ qj^ Qtj^1 bull The only difference between

qtj and q is the presence of D against d is the position of F wi th respect to

the inertial frame FQ whereas d is defined as the offset position of F relative to and

projected on the frame It can be expressed as d = dXidyidZi^ For the

special case of link 1 D = d

From Figure 2-1 can written as

Di = A - l + T i _ i J i _ i + di) + T V i S i - i f t - i + dXi)8i- (262)

Denoting KAj = $j(Zj + dx- ) Eq(262) can be rewritten in summation form as

i ampi = H ( T i - 1 ^ + T i - l K A i - l - l + T j - l ^ - l ) bull ( 2 - 6 3 )

3 = 1

Introducing the index substitution k mdash j mdash 1 into Eq(263)

i-1 D = ^2 (Tfc_ilt4 + TkKAk6k + Tkilk) + T^di (264)

k=l

since d = D TQ = 1^ and IQ DQ SQ are null vectors Recasting Eq(264) in

matr ix form leads to i-1

Qt Jfe=l

where

and

For the entire system

T fc-1 0 TkKAk

0 0 0 0 0 0 0 0 0 0 0 0 -

r T i - i 0 0 0 0 0 0 0 0 0 G

0 0 0 1

Qt = RpQ

G R n lt x l

(265)

(266)

G Unq x l (267)

(268)

30

where

RP

R 1 0 0 R R 2 0 R RP

2 R 3

0 0 0

R RP R PN _ 1 RN J

(269)

Here RP is the lower block triangular matrix which relates qt and q

233 Velocity transformations

The two sets of velocity coordinates qti and qi are related by the following

transformations

(i) First Transformation

Differentiating Eq(262) with respect to time gives the inertial velocity of the

t h l ink as

D = A - l + Ti_iK + + K A i _ i ^ _ i

+ T j _ i ^ + Ti-ii + T i - i K A i - i ^ + T j - i K A ^ i ^ i (270)

(271) (

Defining K A D = dddXi+1KAi K A L = d d K A and hi = di+1 +lfi +KAî

results in K A ^ i = K A D j i o ^ + K A L i _ i i _ i

= KADi_iiTdi + K A L j _ i ^ _ i

Inserting Eq(271) into Eq(270) and rearranging gives

= A _ + T i ^ d g + K A D ^ x ^ i ^ ^ + Pi-^hi-iH-i

+ T i _ i K A i _ i 5 _ i + T i_ i2 + K A L j _ i 5 j _ i 4 _ i (272)

Using Eq(272) to define recursively - D j _ i and applying the index substitution

c = j mdash 1 as in the case of the position transformation it follows that

Di = ^ T ^ i K D ^ ^ + Pk(hk)ffk + T f c K A f c ^ + TfcKLfc4 Jfc=l (273)

+ T j _ i K D j _ i lt i ii

31

where K D j _ i = + K A D ^ i ^ - i r 7 and K L j = i + K A L j 5 j Finally by following a

procedure similar to that used for the position transformation it can be shown that

for the entire system

t = Rvil (2-74)

where Rv is given by the lower block diagonal matrix

Here

Rv

Rf 0 0 R R$ 0 R Rl RJ

0 0 0

T3V nN-l

R bullN

e Mnqqxl

Ri =

T i _ i K D i _ i TiKAi T j K L j 0 0 0 0 bullo 0 0 0 0 0 0 0

e K n lt x l

RS

T i-1 0 0 o-0 0 0 0 0 0

0 0 0 1

G Mnq x l

(275)

(276)

(277)

(ii) Second Transformation

The second transformation is simply Eq(272) set into matrix form This leads

to the expression for qt as

(278)

where Id Pihi) T j K A j T j K L j 0 0 0 0 0 0

0 0

0 0

0 0

G $lnq x l

Thus for the entire system

(279)

pound = Rnjt + Rdq

= Vdnqq - R nrlRdi (280)

32

where

and

- 0 0 0 RI 0 0

Rn = 0 0

0 0

Rf 0 0 0 rgtd K2 0

Rd = 0 0 Rl

0 0 0

RN-1 0

G 5R n w x l

0 0 0 e 5 R N ^ X L

(281)

(282)

234 Cylindrical coordinate modification

Because of the use of cylindrical coorfmates for the first body it is necessary

to slightly modify the above matrices to account for this coordinate system For the

first link d mdash D = D ^ D S L thus

Qti = 01 =gt DDTigi (283)

where

D D T i

Eq(269) now takes the form

DTL 0 0 0 d 3 0 0 0 ID

0 0 nfj 0

RP =

P f D T T i R2 0 P f D T T i Rp

2 R3

iJfDTTi i^

x l

0 0 0

R PN-I RN

(284)

(285)

In the velocity transformation a similar modification is necessary since D =

DD^DS^ Mak ing the substitution it can be shown that Eq(276) and Eq(279) now

33

takes the form for i = 1

T)V _ pn _ r t j mdash rt^ mdash 0 0 0 0

0 0 0 0 0 0 0 0

(286)

The rest of the matrices for the remaining links ie i = 2 N are unchanged

235 Mass matrix inversion

Returning to the expression for kinetic energy substitution of Eq(274) into

Eq(235) gives the first of two expressions for kinetic energy as

Ke = -q RvT MtRv (287)

Introduction of Eq(280) into Eq(235) results in the second expression for Ke as

1X 2q (hnqq-RnrlRd Mt (idnqq - R nrlRd (288)

Thus there are two factorizations for the systems coupled mass matrix given as

M = RV MtRv

M (hnqq - RnrlRd Mt ( i d n q q - R n r 1 R d

(289)

(290)

Inverting Eq(290)

r-1 _ ( rgtd~^ M 1 = (Ra) l d n q q - Rn)Mf1 (Rd)~l(Idnqq - Rn) (291)

Since both Rd and Mt are block diagonal matrices their inverse is simply the inverse

34

of each block on the diagonal ie

Mr1

h 0 0

0 0

0 0

0 0 0

0

0

0

tN

(292)

Each block has the dimension nq x nq and is independent of the number of bodies

in the system thus the inversion of M is only linearly dependent on N ie an O(N)

operation

236 Specification of the offset position

The derivation of the equations of motion using the O(N) formulation requires

that the offset coordinate d be treated as a generalized coordinate However this

offset may be constrained or controlled to follow a prescribed motion dictated either by

the designer or the controller This is achieved through the introduction of Lagrange

multipliers[51] To begin with the multipliers are assigned to all the d equations

Thus letting f = F - Q Eq(259) becomes

Mq + f = P C A (293)

where A G K 3 - 7 V x l is the vector of Lagrange multipliers and Pc G s R n 9 x 3 A r is the

permutation matrix assigning the appropriate Aj to its corresponding d equation

Inverting M and pre-multiplying both sides by PcT gives

pcTM-pc

= ( dc + PdegTM-^f) (294)

where dc is the desired offset acceleration vector Note both dc and PcTM lf are

known Thus the solution for A has the form

A pcTM-lpc - l dc + PC1 M (295)

35

Now substituting Eq(295) into Eq(293) and rearranging the terms gives

q = S(f Ml) = -M~lf + M~lPc

which is the new constrained vector equation of motion with the offset specified by

dc Note that pre-multiplying M~l by P c T provides the rows of M _ 1 corresponding

to the d equations Similarly post-multiplying by Pc leads the columns of the matrix

corresponding to the d equations

24 Generalized Control Forces

241 Preliminary remarks

The treatment of non-conservative external forces acting on the system is

considered next In reality the system is subjected to a wide variety of environmental

forces including atmospheric drag solar radiation pressure Earth s magnetic field as

well as dissipative devices to mention a few However in the present model only

the effect of active control thrusters and momemtum-wheels is considered These

generalized forces wi l l be used to control the attitude motion of the system

In the derivation of the equations of motion using the Lagrangian procedure

the contribution of each force must be properly distributed among all the generalized

coordinates This is acheived using the relation

nu ^5 Qk=E Pern ^ (2-97)

m=l deg Q k

where Qk and qk are the kth elements of Q and ltf respectively[51] Fem is the

mth external force acting on the system at the location Rm from the inertial frame

and nu is the total number of actuators Derivation of the generalized forces due to

thrusters and momentum wheels is given in the next two subsections

pcTM-lpc (dc + PcTM- Vj (2-96)

36

242 Generalized thruster forces

One set of thrusters is placed on each subsatellite ie al l the rigid bodies

except the first (platform) as shown in Figure 2-3 They are capable of firing in the

three orthogonal directions From the schematic diagram the inertial position of the

thruster located on body 3 is given as

Defining

R3 = di + Tid2 + T2(hi + KA2lt52) + T 3 f c m 3

= d + dfY + T 3 f c m 3

df = Tjdj+i + Tj+1lj+ii + KAj+i5j+i)

the thruster position for body 5 is given as

(298)

(299)

R5 = di+ dfx + d 3 + T5fcm

Thus in general the inertial position for the thruster on the t h body can be given

(2100)

as i - 2

k=l i1 cm (2101)

Let the column matrix

[Q i iT 9 R-K=rR dqj lA n i

WjRi A R i (2102)

where i = 3 5 N and j mdash 1 2 i The vector derivative of R wi th respect to

the scaler components qk is stored in the kth column of Thus for the case of

three bodies Eq(297) for thrusters becomes

1 Q

Q3

3 -1

T 3 T t 2 (2103)

37

Satellite 3

mdash gt -raquo cm

D1=d1

mdash gt

a mdash gt

o

Figure 2-3 Inertial position of subsatellite thruster forces

38

where ft = 0TtaTt^T is the thrust vector for the t h l ink (tether i-1) Here

Ttn and Tta are the thrust forces along the inplane m and out-of-plane Z directions i Pi

respectively For the case of 5-bodies

Q

~Qgt1

0

0

( T ^ 2 ) (2104)

The result for seven or more bodies follows the pattern established by Eq(2104)

243 Generalized momentum gyro torques

When deriving the generalized moments arising from the momemtum-wheels

on each rigid body (satellite) including the platform it is easier to view the torques as

coupled forces From Figure 2-4 it is apparent that for the ith l ink the generalized

forces constituting the couple are

(2105)

where

Fei = Faij acting at ei = eXi

Fe2 = -Fpk acting at e 2 = C a ^ l

F e 3 = F7ik acting at e 3 = eyj

Expanding Eq(2105) it becomes clear that

3

Qk = ^ F e i

i=l

d

dqk (2106)

which is independent of Ri- Thus the moments Mma = 2FaieXi Mm^ = 2FpeXi

and Mmi = 2FlieVi Transforming the coordinates to the body fixed frame using

the rotation matrix the generalized force due to the momentum-wheels on link i can

39

Figure 2-4 Coupled force representation of a momentum-wheel on a rigid body

40

be written as

d d d Qi = Td bull ^ T i t M m a - Tik bull mdashT^Mmp + T ^ bull T J j M ^

QiMmi

where Mm = Mma M m Mmi T

lt3 T J ^ T - T i f c bull ^ T i pound T ^ - ^ T z j

(2107)

(2108)

Note combining the thruster forces and momentum-wheel torques a compact

expression for the generalized force vector for 5 bodies can be written as

Q

Q T 3 0 Q T 5 0

0 0 0

0 lt33T3 Qyen Q5

3T5 0

0 0 0 Q4T5 0

0 0 0 Q5

5T5

M M I

Ti 2 m 3

Quu (2109)

The pattern of Eq(2109) is retained for the case of N links

41

3 C O M P U T E R I M P L E M E N T A T I O N


The governing equations of motion for the TV-body tethered system were deshy

rived in the last chapter using an efficient O(N) factorization method These equashy

tions are highly nonlinear nonautonomous and coupled In order to predict and

appreciate the character of the system response under a wide variety of disturbances

it is necessary to solve this system of differential equations Although the closed-

form solutions to various simplified models can be obtained using well-known analytic

methods the complete nonlinear response of the coupled system can only be obtained

numerically

However the numerical solution of these complicated equations is not a straightshy

forward task To begin with the equations are stiff[52] be they have attitude and

vibration response frequencies separated by about an order of magnitude or more

which makes their numerical integration suceptible to accuracy errors i f a properly

designed integration routine is not used

Secondly the factorization algorithm that ensures efficient inversion of the

mass matrix as well as the time-varying offset and tether length significantly increase

the size of the resulting simulation code to well over 10000 lines This raises computer

memory issues which must be addressed in order to properly design a single program

that is capable of simulating a wide range of configurations and mission profiles It

is further desired that the program be modular in character to accomodate design

variations with ease and efficiency

This chapter begins with a discussion on issues of numerical and symbolic in-

42

tegration This is followed by an introduction to the program structure Final ly the

validity of the computer program is established using two methods namely the conshy

servation of total energy and comparison of the response results for simple particular

cases reported in the literature

32 Numerical Implementation

321 Integration routine

A computer program coded in F O R T R A N was written to integrate the equashy

tions of motion of the system The widely available routine used to integrate the difshy

ferential equations D G E A R [ 5 3 ] is based on Gears predictor-corrector method[54]

It is well suited to stiff differential equations as i t provides automatic step-size adjustshy

ment and error-checking capabilities at each iteration cycle These features provide

superior performance over traditional 4^ order Runge-Kut ta methods

Like most other routines the method requires that the differential equations be

transformed into a first order state space representation This is easily accomplished

by letting

thus

X - ^ ^ f ^ ) =ltbullbull) (3 -2 )

where S(fMdc) is defined by Eq(296) Eq(32) represents the new set of 2nqq

first order differential equations of motion

322 Program structure

A flow chart representing the computer programs structure is presented in

Figure 3-1 The M A I N program is responsible for calling the integration routine It

also coordinates the flow of information throughout the program The first subroutine

43

called is INIT which initializes the system variables such as the constant parameters

(mass density inertia etc) and the ini t ial conditions I C (attitude angles orbital

motion tether elastic deformation etc) from user-defined data files Furthermore all

the necessary parameters required by the integration subroutine D G E A R (step-size

tolerance) are provided by INIT The results of the simulation ie time history of the

generalized coordinates are then passed on to the O U T P U T subroutine which sorts

the information into different output files to be subsequently read in by the auxiliary

plott ing package

The majority of the simulation running time is spent in the F C N subroutine

which is called repeatedly by D G E A R F C N generates the fv(Xi) vector defined in

Eq(32) It is composed of several subroutines which are responsible for the comshy

putation of all the time-varying matrices vectors and scaler variables derived in

Chapter 2 that comprise the governing equations of motion for the entire system

These subroutines have been carefully constructed to ensure maximum efficiency in

computational performance as well as memory allocation

Two important aspects in the programming of F C N should be outlined at this

point As mentioned earlier some of the integrals defined in Chapter 2 are notably

more difficult to evaluate manually Traditionally they have been integrated numershy

ically on-line (during the simulation) In some cases particularly stationkeeping the

computational cost is quite reasonable since the modal integrals need only be evalushy

ated once However for the case of deployment or retrieval where the mode shape

functions vary they have to be integrated at each time-step This has considerable

repercussions on the total simulation time Thus in this program the integrals are

evaluated symbolically using M A P L E V[55] Once generated they are translated into

F O R T R A N source code and stored in I N C L U D E files to be read by the compiler

Al though these files can be lengthy (1000 lines or more) for a large number of flexible

44

INIT

DGEAR PARAM

IC amp SYS PARAM

MAIN

lt

DGEAR

lt

FCN

OUTPUT

STATE ACTUATOR FORCES

MOMENTS amp THRUST FORCES

VIBRATION

r OFFS DYNA

ET MICS

Figure 3-1 Flowchart showing the computer program structure

45

modes they st i l l require less time to compute compared to their on-line evaluation

For the case of deployment the performance improvement was found be as high as

10 times for certain cases

Secondly the O(N) algorithm presented in the last section involves matrices

that contained mostly zero terms (RP Rv and Mplusmn) The sparse structure of these mashy

trices can be used advantageously to improve the computational efficiency by avoiding

multiplications of zero elements The result is a quick evaluation of fv(Xt) Moreshy

over there is a considerable saving of computer memory as zero elements are no

longer stored

Final ly the C O N T R O L subroutine which is part of F C N is designed to calshy

culate the appropriate control forces torques and offset dynamics which are required

to stabilize the librational ( A T T I T U D E ) and vibrational ( V I B R A T I O N ) motion of

the system

33 Verification of the Code

The simulation program was checked for its validity using two methods The

first verified the conservation of total energy in the absence of dissipation The second

approach was a direct comparison of the planar response generated by the simulation

program with those available in the literature A s numerical results for the flexible

three-dimensional A^-body model are not available one is forced to be content with

the validation using a few simplified cases

331 Energy conservation

The configuration considered here is that of a 3-body tethered system with

the following parameters for the platform subsatellite and tether

46

h =

1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760

kg bull m 2 platform inertia

kg bull m 2 end-satellite inertia 200 0 0

0 400 0 0 0 400

m = 90000 kg mass of the space station platform

m2 = 500 kg mass of the end-satellite

EfAt = 61645 N tether elastic stiffness

pt mdash 49 kg km tether linear density

The tether length is taken to remain constant at 10 km (stationkeeping case)

A n in i t ia l disturbance in pitch and roll of 2deg and 1deg respectively is given to both

the rigid bodies and the flexible tether In addition the tether is ini t ia l ly deflected

by 45 m in the longitudinal direction at its end and 05 m in both the inplane and

out-of-plane directions in the first mode The tether attachment point offset at the

platform (satellite 1) end is 1 m in all the three directions ie d2 = 11 lTm The

attachment point at the subsatellite (satellite 2) end is 1 m in the x direction ie

fcm3 = 10 0T

The variation in the kinetic and potential energies is presented for the above-

mentioned case in Figure 3-2(a) As seen from the plot there is a continuous mutual

exchange between the kinetic and potential energies however the variation in the

total energy remains zero (Figure 3-2b) It should be noted that after 1 orbit the

variation of both kinetic and potential energy does not return to 0 This is due to

the attitude and elastic motion of the system which shifts the centre of mass of the

tethered system in a non-Keplerian orbit

Similar results are presented for the 5-body case ie a double pendulum (Figshy

ure 3-3) Here the system is composed of a platform and two subsatellites connected

47

STSS configuration 3-Body a 1(0)=a 2(0)=a t(0)=2deg P1(0)=p2(0)=p(0)=1deg 5x(0)=45m 8y(0)=52(0)=05m dx(0)=dy(0)=d2(0)=1m Stationkeeping lt=10km

Variation of Kinetic and Potential Energy

(a) Time (Orbits) Percent Variation in Total Energy

OOEOh

-50E-12

-10E-11 I i i

0 1 2 (b) Time (Orbits)

Figure 3-2 Kinetic and potential energy transfer for the three-body platform based tethered satellite system (a) variation of kinetic and potential energy (b) percent change in total energy of system

48

STSS configuration 5-Body a 1(0)=a 2(0)=a t 1(0)=2deg a 3(0)=a t 2(0)=25deg P1(0)=p2(0)=p3(0)=pt1(0)=pt2(0)=1lt

5x(0)=5m 8y(0)=5z(0)=05m dx(0)=dy(0)=dz(0)=0 Stationkeeping 1 =12=1 Okm

(a)

20E6

10E6r-

00E0

-10E6

-20E6


-

1 I I I 1 I I

I I

AP e ~_

i i 1 i i i i

00E0

UJ -50E-12 U J

lt

Time (Orbits) Percent Variation in Total Energy

-10E-11h

(b) Time (Orbits)

Figure 3 -3 Kinet ic and potential energy transfer for the five-body platform based tethered satellite system (a) variation of kinetic and potential enshyergy (b) percent change in total energy of system

49

in sequence by two tethers The two subsatellites have the same mass and inertia

ie 7713 = 7 7 i 2 a n d I3 = I21 a n d are separated by identical tether segments each 10

k m in length The platform has the same properties as before however there is no

tether attachment point offset present in the system (d = fcmi = 0)

In al l the cases considered energy was found to be conserved However it

should be noted that for the cases where the tether structural damping deployment

retrieval attitude and vibration control are present energy is no longer conserved

since al l these features add or subtract energy from the system

332 Comparison with available data

The alternative approach used to validate the numerical program involves

comparison of system responses with those presented in the literature A sample case

considered here is the planar system whose parameters are the same as those given

in Section 331 A n ini t ia l pitch disturbance of 2deg is imparted to both the platform

and the tether together with an ini t ial tether deformation of 08 m and 001 m in

the longitudinal and transverse directions respectively in the same manner as before

The tether length is held constant at 5 km and it is assumed to be connected to the

centre of mass of the rigid platform and subsatellite The response time histories from

Ref[43] are compared with those obtained using the present program in Figures 3-4

and 3-5 Here ap and at represent the platform and tether pitch respectively and B i

C i are the tethers longitudinal and transverse generalized coordinates respectively

A comparison of Figures 3-4 and 3-5 clearly shows a remarkable correlation

between the two sets of results in both the amplitude and frequency The same

trend persisted even with several other comparisons Thus a considerable level of

confidence is provided in the simulation program as a tool to explore the dynamics

and control of flexible multibody tethered systems

50

a p(0) = 2deg 0(0) = 2C

6(0) = 08 m

0(0) = 001 m

Stationkeeping L = 5 km

D p y = 0 D p 2 = 0

a oo

c -ooo -0 01

Figure 3 - 4 Simulation results for the platform based three-body tethered system originally presented in Ref[43]

51

STSS configuration 3-Body a1(0)=a2(0)=at(0)=2deg p1(0)=(32(0)=(3t(0)=0 8x(0)=08m 5y(0)=001m 5Z(0)=0 dx(0)=dy(0)=dz(0)=0 Stationkeeping l=5km

Satellite 1 Pitch Angle Tether Pitch Angle

i -i i- i i i i i i d

1 2 0 1 2 Time (Orbits) Time (Orbits)

Longitudinal Vibration

^MAAAAAAAWVWWWWVVVW

V bull i bull i I i J 00 05 10 15 20

Time (Orbits) Transverse Vibration

Time (Orbits)

Figure 3-5 Simulation results for the platform based three-body tethered system obtained using the present computer program

52

4 D Y N A M I C SIMULATION


Understanding the dynamics of a system is critical to its design and develshy

opment for engineering application Due to obvious limitations imposed by flight

tests and simulation of environmental effects in ground based facilities space based

systems are routinely designed through the use of numerical models This Chapter

studies the dynamical response of two different tethered systems during deployment

retrieval and stationkeeping phases In the first case the Space platform based Tethshy

ered Satellite System (STSS) which consists of a large mass (platform) connected to

a relatively smaller mass (subsatellite) with a long flexible tether is considered The

other system is the O E D I P U S B I C E P S configuration involving two comparable mass

satellites interconnected by a flexible tether In addition the mission requirement of

spin about an arbitrary axis the tether axis for O E D I P U S - A C and cartwheeling or

spin about the orbit normal for the proposed B I C E P S mission is accounted for

A s mentioned in Chapter 2 the tether flexibility is modelled using the asshy

sumed mode discretization method Although the simulation program can account

for an arbitrary number of vibrational modes in each direction only the first mode

is considered in this study as it accounts for most of the strain energy[34] and hence

dominates the vibratory motion The parametric study considers single as well as

double pendulum type systems with offset of the tether attachment points The

systems stability is also discussed

42 Parameter and Response Variable Definitions

The system parameters used in the simulation unless otherwise stated are

53

selected as follows for the STSS configuration

1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760

h =

h = 200 0 0

0 400 0 0 0 400

bull m = 90000 kg (mass of the space platform)

kg bull m 2 (platform inertia) 760 J

kg bull m 2 (subsatellite inertia)

bull 7772 = 500 kg (mass of the subsatellite)

bull EtAt = 61645 N (tether stiffness)

bull Pt = 49 k g k m (tether density)

bull Vd mdash 05 (tether structural damping coefficient)

bull fcm^ = l 0 0 r m (tether attachment point at the subsatellite)

The response variables are defined as follows

bull ai0 satellite 1 (platform) pitch and roll angles respectively

bull CK2 2 satellite 2 (subsatellite) pitch and roll angles respectively

bull at fit- tether pitch and roll angles respectively

bull If tether length

bull d = dx dy dzT- tether attachment position relative to satellite 1 (platform)

mdash

bull S = 8X 8y Sz tether flexible generalized coordinates in the longitudinal x

inplane transverse y and out-of-plane transverse z directions respectively

The attitude angles and are measured with respect to the Local Ve r t i c a l -

Loca l Horizontal ( L V L H ) frame A schematic diagram illustrating these variable

is presented in Figure 4-1 The system is taken to be in a nominal circular orbit at

an altitude of 289 km with an orbital period of 903 minutes

54

Satellite 1 (Platform) Y a w

Figure 4-1 Schematic diagram showing the generalized coordinates used to deshyscribe the system dynamics

55

43 Stationkeeping Profile

To facilitate comparison of the simulation results a reference case based on

the three-body STSS system with zero offset (d = 0 r c m 3 = 0) and a tether length

of It = 20 km is considered first (Figure 4-2) The system is subjected to an ini t ial

disturbance in pitch and roll of 2deg and 1deg respectively to all the three bodies

In addition an ini t ia l longitudinal deflection of 14 m from the tethers unstretched

position is given together with a transverse inplane and out-of-plane deflection of 1

m at the tethers mid-length From the attitude response given in Figure 4-2(a) it is

apparent that the three bodies oscillate about their respective equilibrium positions

Since coupling between the individual rigid-body dynamics is absent (zero offset) the

corresponding librational frequencies are unaffected Satellite 1 (Platform) displays

amplitude modulation arising from the products of inertia The result is a slight

increase of amplitude in the pitch direction and a significantly larger decrease in

amplitude in the roll angle

Figure 4-2(b) clearly shows coupling of the tethers attitude motion with its

flexibility dynamics The tether vibrates in the axial direction about its equilibrium

position (at approximately 138 m) with two vibration frequencies namely 012 Hz

and 31 x 1 0 - 4 Hz The former is the first longitudinal flexible mode which is dissishy

pated through the structural damping while the latter arises from the coupling with

the pitch motion of the tether Similarly in the transverse direction there are two

frequencies of oscillations arising from the flexible mode and its coupling wi th the

attitude motion As expected there is no apparent dissipation in the transverse flexshy

ible motion This is attributed to the weak coupling between the longitudinal and

transverse modes of vibration since the transverse strain in only a second order effect

relative to the longitudinal strain where the dissipation mechanism is in effect Thus

there is very litt le transfer of energy between the transverse and longitudinal modes

resulting in a very long decay time for the transverse vibrations

56

STSS configuration 3-Body a1(0)=a2(0)=a(0)=2deg (31(0)=p2(0)=p(0)=1deg 8x(0)=14m8y(0)=8z(0)=1m dx(0)=dy(0)=dz(0)=0 Stationkeeping lt=20km

Satellite 1 Pitch Angle Satellite 1 Roll Angle

0 1 2 3 4 5 0 1 2 3 4 5 Time (Orbits) Time (Orbits)


(a) Time (Orbits) Time (Orbits)

Figure 4-2 Stationkeeping dynamics of the three-body STSS configuration withshyout offset (a) attitude response

57

STSS configuration 3-Body a1(0)=cx2(0)=cx(0)=2deg P1(0)=p2(0)=p(0)=1deg 5x(0)=14m8y(0)=82(0)=1m dx(0)=dy(0)=dz(0)=0 Stationkeeping lt=20km

Tether Longitudinal Vibration 16| 1 1 1 1 1 r-

Time (Orbits) Tether In-Plane Transverse Vibration

imdash 1 mdash 1 mdash 1 mdash 1 mdashr

r I I 1 1 1 I 1 1 1 1 I 1 bull ^ bull bull I I J 0 1 2 3 4 5

Time (Orbits) Tether Out-of-Plane Transverse Vibration

(b) Time (Orbits)

Figure 4-2 Stationkeeping dynamics of the three-body STSS configuration withshyout offset (b) vibration response

58

Introduction of the attachment point offset significantly alters the response of

the attitude motion for the rigid bodies W i t h a i m offset along the local vertical

at the platform (dx = 1 m) and fcm^ = 100 m coupling between the tether

and end-bodies is established by providing a lever arm from which the tether is able

to exert a torque that affects the rotational motion of the rigid end-bodies (Figure

4-3) Bo th the platform and subsatellite now oscillate at the same frequency as the

tether whose motion remains unaltered In the case of the platform there is also an

amplitude modulation due to its non-zero products of inertia as explained before

However the elastic vibration response of the tether remains essentially unaffected

by the coupling

Providing an offset along the local horizontal direction (dy = 1 m) results in

a more dramatic effect on the pitch and roll response of the platform as shown in

Figure 4-4(a) Now the platform oscillates about its new equilibrium position of

-90deg The roll motion is also significantly disturbed resulting in an increase to over

10deg in amplitude On the other hand the rigid body dynamics of the tether and the

subsatellite as well as the tether flexible motion remain the same (Figure 4-4b)

Figure 4-5 presents the response when the offset of 1 m at the platform end is

along the z direction ie normal to the orbital plane The result is a larger amplitude

pitch oscillation about the reference equilibrium position as shown in Figure 4-5(a)

while the roll equilibrium is now shifted to approximately 90deg Note there is little

change in the attitude motion of the tether and end-satellites however there is a

noticeable change in the out-of-plane transverse vibration of the tether (Figure 4-5b)

which may be due to the large amplitude platform roll dynamics

Final ly by setting a i m offset in all the three direction simultaneously the

equil ibrium position of the platform in the pitch and roll angle is altered by approxishy

mately 30deg (Figure 4-6a) However the response of the system follows essentially the

59

STSS configuration 3-Body a1(0)=cx2(0)=cxt(0)=2deg p1(0)=(32(0)=pt(0)=1deg 8x(0)=14m8y(0)=8z(0)=1m dx(0)=1m dy(0)=dz(0)=0 Stationkeeping lt=20km


Time (Orbits) Time (Orbits) Tether Pitch Angle Tether Roll Angle

L bull I I bull bull lt bull bull bull raquo bull bull bull bull -I h I I I I i H 0 1 2 3 4 5 0 1 2 3 4 5

Time (Orbits) Time (Orbits) Satellite 2 Pitch Angle Satellite 2 Roll Angle


Figure 4-3 Stationkeeping dynamics of the three-body STSS configuration with offset along the local vertical (a) attitude response

60

STSS configuration 3-Body a 1(0)=a 2(0)=a t(0)=2deg (31(0)=p2(0)=pt(0)=1deg 5x(0)=14m5y(0)=5z(0)=1m dx(0)=1mdy(0)=dz(0)=0 Stationkeeping lt=20km

Tether Longitudinal Vibration 1 6 i mdash i mdash i mdash i mdash i mdash i mdash i mdash i mdash mdash i mdash i mdash mdash mdash mdash mdash i mdash mdash lt -

Time (Orbits)

Tether In-Plane Transverse Vibration

y bull i i i i i i i i i i i i i i i i i 0 1 2 3 4 5

Time (Orbits)

Tether Out-of-Plane Transverse Vibration r 1 1 1 1 1 1 1 1 1 1 1 1 1 1 q

h i I i i I i i i i I i i i i 3

0 1 2 3 4 5 (b) Time (Orbits)

Figure 4-3 Stationkeeping dynamics of the three-body STSS configuration with offset along the local vertical (b) vibration response

61

STSS configuration 3-Body a1(0)=cx2(0)=o(0)=2deg (31(0)=p2(0)=p(0)=1deg 8x(0)=14m8y(0)=8z(0)=1m dx(0)=dz(0)=0 dy(0)=1m Stationkeeping lt=20km

Satellite 1 Pitch Angle i 1 i

Satellite 1 Roll Angle

1 2 3 4 Time (Orbits)

Tether Pitch Angle


Tether Roll Angle

CD

edT

1 2 3 Time (Orbits)

Satellite 2 Pitch Angle



CD

C O

(a)

Figure 4-



4 Stationkeeping dynamics of the three-body STSS configuration with offset along the local horizontal (a) attitude response

62

STSS configuration 3-Body a1(0)=cx2(0)=ot(0)=2deg p1(0)=p2(0)=pt(0)=1deg 8x(0)=14m8y(0)=8z(0)=1m dx(0)=dz(0)=0 dy(0)=1m Stationkeeping lt=20km

Tether Longitudinal Vibration 16| 1 gt 1 bull 1 1 1

0 1 2 3 4 5


i mdash 1 mdash 1 mdash 1 mdash 1 mdash i mdash mdash lt mdash 1 mdash 1 mdash i mdash mdash 1 mdash 1 mdash

V I I bull bull bull I i i i I i i 1 1 3

0 1 2 3 4 5 Time (Orbits)

Tether Out-of-Plane Transverse Vibration r 1 1 1 1 1 1 1 1 1 1 1 gt bull | q

V i 1 1 1 1 1 mdash 1 mdash 3

0 1 2 3 4 5 (b) Time (Orbits)

Figure 4-4 Stationkeeping dynamics of the three-body STSS configuration with offset along the local horizontal (b) vibration response

63

STSS configuration 3-Body a1(0)=o2(0)=at(0)=2deg (31(0)=p2(0)=pt(0)=1deg 8x(0)=14m8y(0)=8z(0)=1m dx(0)=dy(0)=0 dz(0)=1m Stationkeeping lt=20km


2 3 4 Time (Orbits)

Tether Pitch Angle


Tether Roll Angle



2 3 4 Time (Orbits)


(a)

Figure 4-


2 3 4 Time (Orbits)

5 Stationkeeping dynamics of the three-body STSS configuration with offset along the orbit normal (a) attitude response

64

STSS configuration 3-Body a1(0)=cx2(0)=ot(0)=2deg P1(0)=p2(0)=(3(0)=1deg 8x(0)=14m8y(0)=82(0)=1m dx(0)=dy(0)=0 d2(0)=1m Stationkeeping lt=20km

Tether Longitudinal Vibration 1 6 i mdash r mdash i mdash mdash i mdash i mdash mdash i mdash i mdash mdash i mdash mdash i mdash 1 mdash 1 mdash i mdash mdash lt mdash lt mdash lt mdash r


(b) Time (Orbits)

Figure 4-5 Stationkeeping dynamics of the three-body STSS configuration with offset along the orbit normal (b) vibration response

65

same trend with only minor perturbations in the transverse elastic vibratory response

of the tether (Figure 4-6b)

44 Tether Deployment

Deployment of the tether from an ini t ial length of 200 m to 20 km is explored

next Here the deployment length profile is critical to ensure the systems stability

It is not desirable to deploy the tether too quickly since that can render the tether

slack Hence the deployment strategy detailed in Section 216 is adopted The tether

is deployed over 35 orbits with the sinusoidal acceleration and deceleration occurring

over a 4 km length ( A ^ 2 ) with the remaining deployment at a constant velocity of

V0 = 146 ms

Initially the tether is stretched only slightly S = 1304 x 1 0 _ 3 0 0 m

with al l the other states of the system ie pitch roll and transverse displacements

remaining zero The response for the zero offset case is presented in Figure 4-7 For

the platform there is no longer any coupling with the tether hence it oscillates about

the equilibrium orientation determined by its inertia matrix However there is st i l l

coupling between the subsatellite and the tether since rcm^ mdash 100 m resulting

in complete domination of the subsatellite dynamics by the tether Consequently

the pitch motion ini t ial ly grows by almost 50deg but eventually subsides as the tether

deployment rate decreases It is interesting to note that the out-of-plane motion is not

affected by the deployment This is because the Coriolis force which is responsible

for the tether pitch motion does not have a component in the z direction (as it does

in the y direction) since it is always perpendicular to the orbital rotation (Q) and

the deployment rate (It) ie Q x It

Similarly there is an increase in the amplitude of the transverse elastic oscilshy

lations of the tether again due to the Coriolis effect (Figure 4-7b) Moreover as the

66

STSS configuration 3-Body a1(0)=a2(0)=cxt(0)=2deg P1(0)=p2(0)=pt(0)=1deg 8x(0)=14m5y(0)=52(0)=1m dx(0)=dy(0)=d2(0)=1m Stationkeeping lt=20km


i i i i i i i i i i i 0 1 2 3 4 5 0 1 2 3 4 5


l i i i i -I h i i i i I i i i i l- i i i i l i i i i I i i- i i H 0 1 2 3 4 5 0 1 2 3 4 5



Figure 4-6 Stationkeeping dynamics of the three-body STSS configuration with offset along the local horizontal local vertical and orbit normal (a) attitude response

67

STSS configuration 3-Body a1(0)=cc2(0)=a(0)=2deg p1(0)=p2(0)=(3t(0)=1deg 5x(0)=14m6y(0)=52(0)=1m dx(0)=dy(0)=d2(0)=1m Stationkeeping lt=20km

Tether Longitudinal Vibration 1 6 1 mdash i mdash i mdash i mdash mdash i mdash 1 mdash 1 mdash mdash mdash i mdash mdash 1 mdash 1 mdash 1 mdash i mdash 1 mdash 1 mdash 1 mdash 1 mdash r

0 1 2 3 4 5


(b) Time (Orbits)

Figure 4-6 Stationkeeping dynamics of the three-body STSS configuration with offset along the local horizontal local vertical and orbit normal (b) vibration response

68

STSS configuration 3-Body a(0)=a2(0)=ot(0)=0 P1(0)=p2(0)=pt(0)=0 8X(0)=1304 x 103m 8y(0)=8z(0)=0 dx(0)=dy(0)=dz(0)=0 Deployment

l=02km to 20km in 35 Orbits

Tether Length Profile

E



Satellite 1 Roll Angle mdash i mdash i mdash i mdash i mdash i mdash i mdash i mdash i mdash [ mdash i mdash i mdash i mdash i mdash | mdash


Tether Pitch Angle


Tether Roll Angle

CD

oo ax

i imdashimdashimdashImdashimdashimdashimdashr-

_ 1 _ L





-50

(a) 1 2 3 4

Time (Orbits) 1 2 3 4

Time (Orbits)

Figure 4-7 Deployment dynamics of the three-body STSS configuration without offset (a) attitude response

69

STSS configuration 3-Body a1(0)=a2(0)=at(0)=0 p1(0)=p2(0)=(3t(0)=0 8X(0)=1304x 103m 8y(0)=52(0)=0 dx(0)=dy(0)=d2(0)=0 Deployment


Tether Longitudinal Vibration

to

-02

(b)

Figure 4-7

2 3 Time (Orbits)


0 1 1 bull 1 1


Tether Out-of-Plane Transverse Vibration

5

1 1 1 1 1 i i | i i | ^

_ i i i i L -I

1 2 3 4 5 Time (Orbits)

Deployment dynamics of the three-body STSS configuration without offset (b) vibration response

70

tether elongates its longitudinal static equilibrium position also changes due to an

increase in the gravity-gradient tether tension It may be pointed out that as in the

case of attitude motion there are no out-of-plane tether vibrations induced

When the same deployment conditions are applied to the case of 1 m offset in

the x direction (Figure 4-8) coupling effects are re-introduced The platform pitch

exceeds -100deg during the peak deployment acceleration However as the tether pitch

motion subsides the platform pitch response returns to a smaller amplitude oscillation

about its nominal equilibrium On the other hand the platform roll motion grows

to over 20deg in amplitude in the terminal stages of deployment Longitudinal and

transverse vibrations of the tether remain virtually unchanged from the zero offset

case except for minute out-of-plane perturbations due to the platform librations in

roll

45 Tether Retrieval

The system response for the case of the tether retrieval from 20 km to 200 m

with an offset of 1 m in the x and z directions at the platform end is presented in

Figure 4-9 Here the Coriolis force induced during retrieval renders the tether unstable

(Figure 4-9a) Consequently the pitch motion of the platform is also destabilized

through coupling The z offset provides an additional moment arm that shifts the

roll equilibrium to 35deg however this degree of freedom is not destabilized From

Figure 4-9(b) it is apparent that the transverse mode of the tether vibration is also

disturbed producing a high frequency response in the final stages of retrieval This

disturbance is responsible for the slight increase in the roll motion for both the tether

and subsatellite

46 Five-Body Tethered System

A five-body chain link system is considered next To facilitate comparison with

71

STSS configuration 3-Body a(0)=a2(0)=at(0)=0 P1(0)=P2(0)=P(0)=0 5X(0)=1304 x 103m 8y(0)=8z(0)=0 dx(0)=1mdy(0)=d2(0)=0 Deployment

lt=02km to 20km in 35 Orbits


E


1 2 3 Time (Orbits)


bull bull bull i bull bull bull bull i


Tether Pitch Angle

1 2 3 Time (Orbits)

Tether Roll Angle

i i i i J I I i _




Satellite 2 Roll Angle 3E-3h

D )

T J ^ 0 E 0 eg

CQ

-3E-3

(a) 1 2 3 4


Time (Orbits)

Figure 4-8 Deployment dynamics of the three-body STSS configuration with offset along the local vertical (a) attitude response

7 2

STSS configuration 3-Body a1(0)=a2(0)=at(0)=0 p1(0)=P2(0)=(3t(0)=0 Sx(0)=1304 x 103m 5y(0)=5z(0)=0 dx(0)=1mdy(0)=dz(0)=0 Deployment




n 1 1 1 f

I I I _j I I I I I 1 1 1 1 1 2 3 4 5


J 1 I I 1 I I I I I I I I L _ _ I I d 1 2 3 4 5

Time (Orbits)

Deployment dynamics of the three-body STSS configuration with offset along the local vertical (b) vibration response

(b)

Figure 4-8

73

STSS configuration 3-Body a1(0)=a2(0)=at(0)=0 p(0)=P2(0)=(3t(0)=0 Sx(0)=14m8y(0)=82(0)=0 dy(0)=0dx(0)=d2(0)=1m Retrieval



(a) Time (Orbits)

Figure 4-9 Retrieval dynamics of the along the local vertical ar


i i i i i i i i i i I 0 1 2 3

Time (Orbits) Satellite 1 Roll Angle

Time (Orbits) Tether Roll Angle

b_ i i I i I i i L _ J


Satellite 2 Roll Angle _ mdash | r i i | 1 i i imdashmdashj i 1 r mdash i mdash

04- -


three-body STSS configuration with offset d orbit normal (a) attitude response

74

STSS configuration 3-Body a1(0)=a2(0)=ocl(0)=0 (31(0)=f32(0)=pt(0)=0 8x(0)=14m8y(0)=6z(0)=0 dy(0)=0dx(0)=dz(0)=1m Retrieval



E 10 to


1 2 Time (Orbits)


(b) Time (Orbits)

Figure 4-9 Retrieval dynamics of the three-body STSS configuration with offset along the local vertical and orbit normal (b) vibration response

75

the three-body system dynamics studied earlier the chain is extended through the

addition of two bodies a tether and a subsatellite with the same physical properties

as before (Section 42) Thus the five-body system consists of a platform (satellite 1)

tether 1 subsatellite 1 (satellite 2) tether 2 and subsatellite 2 (satellite 3) as shown

in Figure 4-10 The offset of tether 1 at the platform-end is denoted by d2 while that

of tether 2 to subsatellite 1 as 04 In this numerical example fcm^ = fcm^ = 0 ie

tethers 1 and 2 are attached to the centre of mass of subsatellites 1 and 2 respectively

The system response for the case where the tether attachment points coincide

wi th the centres of mass of the rigid bodies ie zero offset (d2 = d mdash 0) is presented

in Figure 4-11 Here ogtt and at2 represent the pitch motion of tethers 1 and 2

respectively whereas 03 is the pitch motions of satellite 3 A similar convention

is adopted for the rol l angle 0 A s before the zero offset eliminates the coupling

between the tethers and satellites such that they are now free to librate about their

static equil ibrium positions However their is s t i l l mutual coupling between the two

tethers This coupling is present regardless of the offset position since each tether is

capable of transferring a force to the other The coupling is clearly evident in the pitch

response of the two tethers (Figure 4 - l l a ) O n the other hand the roll motion appears

uncoupled as the in i t ia l conditions in rol l for the two tethers are identical Thus the

motion is in phase and there is no transfer of energy through coupling A s expected

during the elastic response the tethers vibrate about their static equil ibrium positions

and mutually interact (Figure 4 - l l b ) However due to the variation of tension along

the tether (x direction) they do not have the same longitudinal equilibrium Note

relatively large amplitude transverse vibrations are present particularly for tether 1

suggesting strong coupling effects with the longitudinal oscillations

76

Figure 4-10 Schematic diagram of the five-body system used in the numerical example

77

STSS configuration 5-Body a1(0)=(x2(0)=at1(0)=2deg a3(0)=a12(0)=25deg p1(0)=r32(0)=p3(0)=pt1(0)=pt2(0)=r 82(0)=84(0)=505)05m d2(0)=d4(0)=000 Stationkeeping lt1=lu=10km Pitch Roll


o 1 2 Time (Orbits)

Tether 1 Pitch and Roll Angle

0 1 2 Time (Orbits)


Time (Orbits) Satellite 2 Pitch and Roll Angle


Time (Orbits) Time (Orbits)

Figure 4-11 Stationkeeping dynamics of the five-body STSS configuration withshyout offset (a) attitude response

78

STSS configuration 5-Body a1(0)=a2(0)=oct1(0)=2deg a3(0)=at2(0)=25o

p1(0)=p2(0)=(33(0)=pt1(0)=pt2(0)=10

62(0)=54(0)=50505m d2(0)=d4(0)=000 Stationkeeping lt1=1 =1 Okm

Tether 1 Longitudinal Vibration Tether 2 Longitudinal Vibration i i 1 i 1 r 1 1 1 1

(b) Time (Orbits) Time (Orbits)

Figure 4-11 Stationkeeping dynamics of the five-body STSS configuration withshyout offset (b) vibration response

79

The system dynamics during deployment of both the tethers in the double-

pendulum configuration is presented in Figure 4-12 Deployment of each tether takes

place from 200 m to 20 km in 35 orbits The pitch motion of the system is similar

to that observed during the three-body case The Coriolis effect causes the tethers

to pitch through a large angle ( laquo 50deg ) which in turn disturbs the platform and

subsatellite due to the presence of offset along the local vertical On the other hand

the roll response for both the tethers is damped to zero as the tether deploys in

confirmation with the principle of conservation of angular momentum whereas the

platform response in roll increases to around 20deg Subsatellite 2 remains virtually

unaffected by the other links since the tether is attached to its centre of mass thus

eliminating coupling Finally the flexibility response of the two tethers presented in

Figure 4-12(b) shows similarity with the three-body case

47 BICEPS Configuration

The mission profile of the Bl-static Canadian Experiment on Plasmas in Space

(BICEPS) is simulated next It is presently under consideration by the Canadian

Space Agency To be launched by the Black Brant 2000500 rocket it would inshy

volve interesting maneuvers of the payload before it acquires the final operational

configuration As shown in Figure 1-3 at launch the payload (two satellites) with

an undeployed tether is spinning about the orbit normal (phase 1) The internal

dampers next change the motion to a flat spin (phase 2) which in turn provides

momentum for deployment (phase 3) When fully deployed the one kilometre long

tether will connect two identical satellites carrying instrumentation video cameras

and transmitters as payload

The system parameters for the BICEPS configuration are summarized below [59 0 0 1

bull I = I2 = 0 366 0 kg bull m 2 (satellite inertias) [ 0 0 392_

bull mi = 7772 = 200 kg (mass of the satellites)

80

STSS configuration 5-Body a(0)=cx2(0)=al1(0)=2deg a3(0)=c (0)=25o

P1(0)=P2(0)=p3(0)=f3t1(0)=Pt2(0)=r 52(0)=84(0)=1304x 10300m d2(0)=d4(0)=1G0m Deployment lt1=le=02km to 20km in 35 Orbits

Pitch Roll Satellite 1 Pitch Angle

Time (Orbits) Satellite 2 Pitch and Roll Angle T 1mdash1 1 1 I I I I | I I I 1 I | I I lt~r

Tether Length Profile i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i

i i i i i 0 1 2 3 4 5



Tether 1 amp 2 Roll Angle L i | I I 1 I 1 1 -I

-Ih I- bull I I i i i i I i i i i 1


Satellite 3 Pitch and Roll Angle i- i 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i

L i i i i i i 1 i i mdash i mdash i I i imdashimdashLJ


Figure 4-12 Deployment dynamics of the five-body S T S S configuration with offshyset along the local vertical (a) attitude response

81

STSS configuration 5-Body a1(0)=a2(0)=oct1(0)=2deg a3(0)=at2(0)=25deg P1(0)=P2(0)=P3(0)=Pt1(0)=pt2(0)=1o

82(0)=54(0)=1304 x 10-300m d2(0)=d4(0)=100m Deployment lt1=l2=02km to 20km in 35 Orbits

Tether 1 Longitudinal Vibration Tether 2 Longitudinal Vibration

Time (Orbits) Time (Orbits) Tether 1 Z Tranverse Vibration Tether 2 Z Tranverse Vibration

Figure 4-12 Deployment dynamics of the five-body STSS configuration with offshyset along the local vertical (b) vibration response

82


bull pt = 30 k g k m (tether density)

bull It = 1 km (tether length)

bull rid = 1-0 (tether structural damping coefficient)

The system is in a circular orbit at a 289 km altitude Offset of the tether

attachment point to the satellites at both ends is taken to be 078 m in the x

direction The response of the system in the stationkeeping phase for a prescribed set

of in i t ia l conditions is given by Figure 4-13 As in the case of the STSS configuration

there is strong coupling between the tether and the satellites rigid body dynamics

causing the latter to follow the attitude of the tether However because of the smaller

inertias of the payloads there are noticeable high frequency modulations arising from

the tether flexibility Response of the tether in the elastic degrees of freedom is

presented in Figure 4-13(b) Note the longitudinal vibrations decay quite rapidly

(lt 05 orbits) due to the structural damping however it has vir tual ly no effect on

the transverse oscillations

The unique mission requirement of B I C E P S is the proposed use of its ini t ia l

angular momentum in the cartwheeling mode to aid in the deployment of the tethered

system The maneuver is considered next The response of the system during this

maneuver wi th an ini t ia l cartwheeling rate of 5 deg s is illustrated in Figure 4-14(a) The

in i t ia l tether length is taken to be 10 m and is deployed to 1 km There is an in i t ia l

increase in the pitch motion of the tether however due to the conservation of angular

momentum the cartwheeling rate decreases proportionally to the square of the change

in tether deployment rate The result is a rapid drop in the cartwheeling rate unti l

the system stops rotating entirely and simply oscillates about its new equilibrium

point Consequently through coupling the end-bodies follow the same trend The

roll response also subsides once the cartwheeling motion ceases However

83

BICEPS configuration 3-Body a1(0)=o2(0)=at(0)=2deg P1(0)=P2(0)=pt(0)=1deg 8X(0)=001 m 6y(0)=52(0)=01 m dx(0)=078m dy(0)=d2(0)=0 Stationkeeping l=1 km

Satellite 1 Pitch Angle Satellite 1 Roll Angle - i 1 1 1 1 i - i q r 1 1 -gt

I- I i t i i i i I H 0 1 2 0 1 2


F mdash 1 mdash 1 mdash mdash 1 mdash i mdash 1 mdash 1 mdash mdash 1 mdash = i F mdash 1 mdash mdash mdash 1 mdash i mdash 1 mdash 1 mdash 1 mdash 1 mdash q


Figure 4-13 Stationkeeping dynamics of the three-body BICEPS configuration with offset along the local vertical (a) attitude response

84

BICEPS configuration 3-Body a1(0)=o2(0)=at(0)=2deg Pl(0)=p2(0)=(3t(0)=1deg 8X(0)=001 m 8y(0)=82(0)=01 m dx(0)=078m dy(0)=d2(0)=0 Stationkeeping lt=1km

Tether Longitudinal Vibration 1E-2[

laquo = 5 E - 3 |

000 002



Time (Orbits)

Figure 4-13 Stationkeeping dynamics of the three-body BICEPS configuration with offset along the local vertical (b) vibration response

85

BICEPS configuration 3-Body a1(0)=o2(0)=at(0)=2deg a1(0)=a2(0)=at(0)=57s Pl(0)=P2(0)=(3t(0)=1deg 8X(0)=115 x 103m 8y(0)=82(0)=0 dx(0)=078m dy(0)=d2(0)=0 Cartwheeling Deployment

lt=10m to 1km in 1 Orbit


E

2000

cu T3

2000

co CD

2000

C D CD

T J


1 2 Time (Orbits)


cn CD

33 cdl

Time (Orbits) Tether Pitch Angle


Time (Orbits) Satellite 2 Pitch Angle

1 2 Time (Orbits)



Figure 4-14 Cartwheeling dynamics with deployment for the three-body BICEPS with offset along the local vertical (a) attitude response

86

BICEPS configuration 3-Body a1(0)=a2(0)=at(0)=2deg a (0)=a2(0)=a(0)=57s P1(0)=P2(0)=f3(0)=1deg 8X(0)=115x103m 8y(0)=8z(0)=0 dx(0)=078m dy(0)=d2(0)=0 Cartwheeling Deployment

l=1 Om to 1 km in 1 Orbit

5E-3h

co

0E0

025

000 E to

-025

002

-002

(b)

Figure 4-14




Time (Orbits)

Cartwheeling dynamics with deployment for the three-body BICEPS with offset along the local vertical (b) vibration response

87

as the deployment maneuver is completed the satellites continue to exhibit a small

amplitude low frequency roll response with small period perturbations induced by

the tether elastic oscillations superposed on it (Figure 4-14b)

48 OEDIPUS Spinning Configuration

The mission entitled Observation of Electrified Distributions of Ionospheric

Plasmas - A Unique Strategy ( O E D I P U S ) also consists of a 1 k m tethered system with

two similar end-masses It was first flown in a suborbital flight in 1989 ( O E D I P U S A )

followed by a recent study in November 1995 ( O E D I P U S - C ) in which the University

of Br i t i sh Columbia was one of the participants Dynamically O E D I P U S represents a

unique system never encountered before Launched by a Black Brant rocket developed

at Br is to l Aerospace L t d the system ie the end-bodies together wi th the tether

spins about the longitudinal axis of the tether to achieve stabilized alignment wi th

Earth s magnetic field

The response of the system undergoing a spin rate of j = l deg s is presented

in Figure 4-15 using the same parameters as those outlined for the B I C E P S conshy

figuration in Section 47 Again there is strong coupling between the three bodies

(two satellites and tether) due to the nonzero offset The spin motion introduces an

additional frequency component in the tethers elastic response which is transferred

to the l ibrational motion of the satellites A s the spin rate increases to 1 0 deg s (Figure

4-16) the amplitude and frequency of the perturbations also increase however the

general character of the response remains essentially the same

88

OEDIPUS configuration 3-Body a1(0)=a2(0)=at(0)=2deg i(0)^[2(0H(0)=Ys p1(0)=p2(0)=p(0)=r 8X(0)=12 x 103m 8y(0)=82(0)=01m dx(0)=078m dy(0)=d2(0)=0 Stationkeeping l=1 km

Satellite 1 Pitch Angle Satellite 1 Roll Angle mdash i 1 1 1 a r 1 1 1 r


Figure 4-15 Spin dynamics (7 = ldegs) of the three-body OEDIPUS configuration with offset along the local vertical (a) attitude response

89

OEDIPUS configuration 3-Body a1(0)=a2(0)=cct(0)=2deg Y1(0)=Y2(0)=Yt(0)=l7s (31(0)=p2(0)=pt(0)=1deg 8X(0)=12 x 103m 8y(0)=5z(0)=01 m dx(0)=078m dy(0)=d2(0)=0 Stationkeeping lt=1 km

1E-2

to

OEO

E 00

to


005



Time (Orbits)

Figure 4-15 Spin dynamics (7 = l deg s ) of the three-body O E D I P U S configuration with offset along the local vertical (b) vibration response

90

OEDIPUS configuration 3-Body a1(0)=a2(0)=a(0)=2deg Yi(0)=Y2(0)=rt(0)=107s 81(0)=p2(0)=pt(0)=1deg 8X(0)=12 x 103m 8y(0)=82(0)=01m dx(0)=078m dy(0)=d2(0)=0 Stationkeeping lt=1km



Figure 4-16 Spin dynamics (7= 10degs) of the three-body OEDIPUS configura tion with offset along the local vertical (a) attitude response

91

OEDIPUS configuration 3-Body a1(0)=a2(0)=ot(0)=2deg Yi(0)H(0)4(0)laquo107s P1(0)=P2(0)=pt(0)=1deg 8X(0)=12 x 103m 8y(0)=82(0)=01m dx(0)=078m dy(0)=dz(0)=0 Stationkeeping lt=1 km


Figure 4-16 Spin dynamics (7 = 10degs) of the three-body OEDIPUS configurashytion with offset along the local vertical (b) vibration response

92

5 ATTITUDE AND VIBRATION CONTROL

51 Att itude Control


The instability in the pitch and roll motions during the retrieval of the tether

the large amplitude librations during its deployment and the marginal stability during

the stationkeeping phase suggest that some form of active control is necessary to

satisfy the mission requirements This section focuses on the design of an attitude

controller with the objective to regulate the librational dynamics of the system

As discussed in Chapter 1 a number of methods are available to accomplish

this objective This includes a wide variety of linear and nonlinear control strategies

which can be applied in conjuction with tension thrusters offset momentum-wheels

etc and their hybrid combinations One may consider a finite number of Linear T ime

Invariant (LTI) controllers scheduled discretely over different system configurations

ie gain scheduling[43] This would be relevant during deployment and retrieval of

the tether where the configuration is changing with time A n alternative may be

to use a Linear T ime Varying ( L T V ) model in conjuction with an adaptive control

scheme where on-line parametric identification may be used to advantage[56] The

options are vir tually limitless

O f course the choice of control algorithms is governed by several important

factors effective for time-varying configurations computationally efficient for realshy

time implementation and simple in character Here the thrustermomentum-wheel

system in conjuction with the Feedback Linearization Technique ( F L T ) is chosen as

it accounts for the complete nonlinear dynamics of the system and promises to have

good overall performance over a wide range of tether lengths It is well suited for

93

highly time-varying systems whose dynamics can be modelled accurately as in the

present case

The proposed control method utilizes the thrusters located on each rigid satelshy

lite excluding the first one (platform) to regulate the pitch and rol l motion of the

tether to which it is attached ie the thrusters located on satellite 2 (link 3) regshy

ulates the attitude motion of tether 1 (link 2) O n the other hand the l ibrational

motion of the rigid bodies (platform and subsatellite) is controlled using a set of three

momentum wheels placed mutually perpendicular to each other

The F L T method is based on the transformation of the nonlinear time-varying

governing equations into a L T I system using a nonlinear time-varying feedback Deshy

tailed mathematical background to the method and associated design procedure are

discussed by several authors[435758] The resulting L T I system can be regulated

using any of the numerous linear control algorithms available in the literature In the

present study a simple P D controller is adopted arid is found to have good perforshy

mance

The choice of an F L T control scheme satisfies one of the criteria mentioned

earlier namely valid over a wide range of tether lengths The question of compushy

tational efficiency of the method wi l l have to be addressed In order to implement

this controller in real-time the computation of the systems inverse dynamics must

be executed quickly Hence a simpler model that performs well is desirable Here

the model based on the rigid system with nonlinear equations of motion is chosen

Of course its validity is assessed using the original nonlinear system that accounts

for the tether flexibility

512 Controller design using Feedback Linearization Technique

The control model used is based on the rigid system with the governing equa-

94

tions of motion given by

Mrqr + fr = Quu (51)

where the left hand side represents inertia and other forces while the right hand side

is the generalized external force due to thrusters and momentum-wheels and is given

by Eq(2109)

Lett ing

~fr = fr~ QuU (52)

and substituting in Eq(296)

ir = s(frMrdc) (53)

Expanding Eq(53) it can be shown that

qr = S(fr Mrdc) - S(QU Mr6)u (5-4)

mdash = Fr - Qru

where S(QU Mr0) is the column matrix given by

S(QuMr6) = [s(Qu(l)Mr0) 5 (Q u ( 2 ) M r | 0 ) S(QU nu) M r | 0 ) ] (55)

and Qu-i) is the i ^ column of Qu Extract ing only the controlled equations from

Eq(54) ie the attitude equations one has

Ire = Frc ~~ Qrcu

(56)

mdash vrci

where vrc is the new control input required to regulate the decoupled linear system

A t this point a simple P D controller can be applied ie

Vrc = qrcd +KvQrcd-Qrc) + Kp(Qrcd~ Qrc)- (57)

Eq(57) represents the secondary controller with Eq(54) as the primary controller

Kp and Kv are the proportional and derivative gain matrices with qrcdi Qrcd

an(^ qrcd

95

as the desired acceleration velocity and position vector for the attitude angles of each

controlled body respectively Solving for u from Eq(56) leads to

u Qrc Frc ~ vrcj bull (58)

5 13 Simulation results

The F L T controller is implemented on the three-body STSS tethered system

The choice of proportional and derivative matrix gains is based on a desired settling

time of ts mdash 05 orbit and a damping factor of = 07 for both the pitch and roll

actuators in each body Given O and ts it can be shown that

-In ) 0 5 V T ^ r (59)

and hence kn bull mdash10

V (510)

where kp and kVj are the diagonal elements of the gain matrices Kp and Kv in

Eq(57) respectively[59]

Note as each body-fixed frame is referred directly to the inertial frame FQ

the nominal pitch equilibrium angle is zero only when measured from the L V L H frame

(OJJ mdash 0) However it is 6 when referred to FQ Hence the desired position vector

qrC(l is set equal to 0(t) ie the time-varying orbital angle for the pitch motion

and zero for the roll and yaw rotations such that

qrcd 37Vxl (511)

96

The desired velocity and acceleration are then given as

e U 3 N x l (512)

E K 3 7 V x l (513)

When the system is in a circular orbit qrc^ = 0

Figure 5-1 presents the controlled response of the STSS system defined in

Chapter 4 in the stationkeeping phase with offset d2 = 111T m and If = 20

km A s mentioned earlier the F L T controller is based on the nonlinear rigid model

Note the pitch and roll angles of the rigid bodies as well as of the tether are now

attenuated in less than 1 orbit (Figure 5-la) Consequently the longitudinal vibration

response of the tether is free of coupling arising from the librational modes of the

tether This leaves only the vibration modes which are eventually damped through

structural damping The pitch and roll dynamics of the platform require relatively

higher control moments Ma and Mpi respectively (Figure 5-lb) since they have to

overcome the extra moments generated by the offset of the tether attachment point

Furthermore the platform demands an additional moment to maintain its orientation

at zero pitch and roll angle since it is not the nominal equilibrium position The nonshy

zero static equilibrium of the platform arises due to its products of inertia On the

other hand the tether pitch and roll dynamics require only a small ini t ia l control

force of about plusmn 1 N which eventually diminishes to zero once the system is

97

and

o 0

Qrcj mdash

o o

0 0

Oiit) 0 0

1

N

N

1deg I o

STSS configuration 3-Body a1(0)=a2(0)=at(0)=2lt

P1(0)=p2(0)=pt(0)=1 8x(0)=14m8y(0)=8z(0)=1m dx(0)=dy(0)=dz(0)=1m Controlled Stationkeeping lt=20km

Tether Pitch and Roll Angles

Sat 1 Pitch and Roll Angles

05 10 15 Time (Orbits)


20

CO CD

T J

cdT

cT

- mdash mdash 1 1 1 1 1 I - 1

Pitch -

A Roll -

A -

1 bull bull

-

1 Time (Orbits)

Tether Y Tranverse Vibration

1 2 Time (Orbits)

Tether Z Tranverse Vibration

gt to N

to

Time (Orbits) Time (Orbits) Longitudinal Vibration

145h

(a) Time (Orbits)

Figure 5-1 Controlled dynamics of the three-body STSS during stationkeeping using the nonlinear rigid FLT controller with offset along the local horizontal local vertical and orbit normal (a) attitude and vibration response

98

1

STSS configuration 3-Body a1(0)=ct2(0)=cct(0)=20

P1(0)=p2(0)=pt(0)=1deg 8x(0)=14m8y(0)=8z(0)=1m dx(0)=dy(0)=d2(0)=1m Controlled Stationkeeping lt=20km

Sat 1 Pitch Control Moment Sat 1 Roll Control Moment

Figure 5-1 Controlled dynamics of the three-body STSS during stationkeeping using the nonlinear rigid FLT controller with offset along the local horizontal local vertical and orbit normal (b) control actuator time histories

99

stabilized Similarly for the end-body a very small control torque is required to

attenuate the pitch and roll response

If the complete nonlinear flexible dynamics model is used in the feedback loop

the response performance was found to be virtually identical with minor differences

as shown in Figure 5-2 For example now the pitch and roll response of the platform

is exactly damped to zero with no steady-state error as against the minor almost

imperceptible deviation for the case of the controller basedon the rigid model (Figure

5-la) In addition the rigid controller introduces additional damping in the transverse

mode of vibration where none is present when the full flexible controller is used

This is due to a high frequency component in the at motion that slowly decays the

transverse motion through coupling

As expected the full nonlinear flexible controller which now accounts the

elastic degrees of freedom in the model introduces larger fluctuations in the control

requirement for each actuator except at the subsatellite which is not coupled to the

tether since f c m 3 = 0 The high frequency variations in the pitch and roll control

moments at the platform are due to longitudinal oscillations of the tether and the

associated changes in the tension Despite neglecting the flexible terms the overall

controlled performance of the system remains quite good Hence the feedback of the

flexible motion is not considered in subsequent analysis

When the tether offset at the platform is restricted to only 1 m along the x

direction a similar response is obtained for the system However in this case the

steady-state error in the platforms rigid body motion is much smaller (Figure 5-3a)

In addition from Figure 5-3(b) the platforms control requirement is significantly

reduced

Figure 5-4 presents the controlled response of the STSS deploying a tether

100

STSS configuration 3-Body a1(0)=a2(0)=oct(0)=2deg p1(0)=P2(0)=p(0)=1deg 8x(0)=14m5y(0)=82(0)=1m dx(0)=dy(0)=dz(0)=1m Controlled Stationkeeping l=20km


c n

T J

cdT

Sat 1 Pitch and Roll Angles 1 r



c n CD

cdT a

o T J

CQ

- - I 1 1 1 I 1 1 1 1

P i t c h

R o l l

bull

1 -

0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


E gt

10 N

CO


145h

(a) Time (Orbits)

Figure 5-2 Controlled dynamics of the three-body STSS during stationkeeping using the nonlinear flexible FLT controller with offset along the local horizontal local vertical and orbit normal (a) attitude and vibration response

101

STSS configuration 3-Body a1(0)=a2(0)=cxt(0)=2deg 31(0)=p2(0)=p(0)=1deg 8x(0)=14m8y(0)=8z(0)=1m dx(0)=dy(0)=d2(0)=1m Controlled Stationkeeping lt=20km

r i i i t i mdash i mdash i mdash i mdash i mdash i mdash i mdash i mdash mdash J 0 1 2 0 1 2

Time (Orbits) Time (Orbits) Tether Pitch Control Thrust Tether Roll Control Thruster


Figure 5-2 Controlled dynamics of the three-body STSS during stationkeeping using the nonlinear flexible FLT controller with offset along the local horizontal local vertical and orbit normal (b) control actuator time histories

102

STSS configuration 3-Body a1(0)=o2(0)=a(0)=2deg P1(0)=p2(0)=(3t(0)=1deg 8x(0)=14m5y(0)=6z(0)=1m dx(0)=1mdy(0)=dz(0)=0 Controlled Stationkeeping lt=20km





CD

eg CQ

0 1 2 Time (Orbits)


1 2 Time (Orbits)


gt CO


145h

(a) Time (Orbits)

Figure 5-3 Controlled dynamics of the three-body STSS during stationkeeping using the nonlinear rigid FLT controller with offset along the local vertical (a) attitude and vibration response

103

STSS configuration 3-Body a1(0)=o2(0)=al(0)=2o

P1(0)=p2(0)=pt(0)=1deg 8x(0)=14m8y(0)=8z(0)=1m dx(0)=1mdy(0)=dz(0)=0 Controlled Stationkeeping lt=20km

Sat 1 Pitch Control Moment 11 1 1 r

Sat 1 Roll Control Moment

0 1 2 Time (Orbits)

Tether Pitch Control Thrust

0 1 2 Time (Orbits)

Sat 2 Pitch Control Moment

0 1 2 Time (Orbits)

Tether Roll Control Thruster

-07 o 1

Time (Orbits Sat 2 Roll Control Moment

1E-5

0E0F

Time (Orbits)

Controlled dynamics of the three-body STSS during stationkeeping using the nonlinear rigid FLT controller with offset along the local vertical (b) control actuator time histories

104

STSS configuration 3-Body a1(0)=a2(0)=a(0)=2deg Pl(0)=p2(0)=pt(0)=1deg 6X(0)=1304 x 103m 8y(0)=82(0)=0 dx(0)=1mdy(0)=d2(0)=0 Controlled Deployment




o 1 2 Time (Orbits)


0 1 2 Time (Orbits)


CD CD

CM CO

Time (Orbits) Tether Z Tranverse Vibration

to

150

100

1 2 3 0 1 2


to

(a)

50

2 3 Time (Orbits)

Figure 5 - 4 Deployment dynamics of the three-body STSS using the nonlinear rigid FLT controller with offset along the local vertical (a) attitude and vibration response

105

STSS configuration 3-Body a1(0)=oc2(0)=al(0)=20

P1(0)=p2(0)=pt(0)=1deg 8X(0)=1304 x 103m 5y(0)=52(0)=0 dx(0)=1mdy(0)=dz(0)=0 Controlled Deployment


Tether Length Profile i 1

2 0 h

E 2L

Sat 1 Pitch Control Moment t mdash r mdash i mdash i mdash | mdash i mdash i mdash i mdash i mdash | mdash r mdash i mdash i mdash i mdash | mdash r mdash

0 5

1 2 3

Time (Orbits) Sat 1 Roll Control Moment

i i t | t mdash t mdash i mdash [ i i mdash r -

1 2

Time (Orbits) Tether Pitch Control Thrust

1 2 3 Time (Orbits)


2 E - 5

1 2 3

Time (Orbits) Sat 2 Pitch Control Moment

1 2


1 E - 5

E

O E O

Time (Orbits)

Deployment dynamics of the three-body STSS using the nonlinear rigid FLT controller with offset along the local vertical (b) control actuator time histories

106

from 02 km to 20 km in 35 orbits with a i m offset along the tether length A s before

the systems attitude motion is well regulated by the controller However the control

cost increases significantly for the pitch motion of the platform and tether due to the

Coriolis force induced by the deployment maneuver (Figure 5-4b) Initially there is a

sinusoidal increase in the pitch control requirement for both the tether and platform

as the tether accelerates to its constant velocity VQ Then the control requirement

remains constant for the platform at about 60 N m as opposed to the tether where the

thrust demand increases linearly Finally when the tether decelerates the actuators

control input reduces sinusoidally back to zero The rest of the control inputs remain

essentially the same as those in the stationkeeping case In fact the tether pitch

control moment is significantly less since the tether is short during the ini t ia l stages

of control However the inplane thruster requirement Tat acts as a disturbance and

causes 6y to nearly double from the uncontrolled value (Figure 4-8a) O n the other

hand the tether has no out-of-plane deflection

Final ly the effectiveness of the F L T controller during the crit ical maneuver of

retrieval from 20 km to 02 k m in 35 orbits is assessed in Figure 5-5 A s in the earlier

cases the system response in pitch and roll is acceptable however the controller is

unable to suppress the high frequency elastic oscillations induced in the tether by the

retrieval O f course this is expected as there is no active control of the elastic degrees

of freedom However the control of the tether vibrations is discussed in detail in

Section 52 The pitch control requirement follows a similar trend in magnitude as

in the case of deployment with only minor differences due to the addition of offset in

the out-of-plane direction ( d 2 = 1 0 1 ^ m) This is also responsible for the higher

roll moment requirement for the platform control (Figure 5-5b)

107

STSS configuration 3-Body a1(0)=X2(0)=a(0)=2deg P1(0)=p2(0)=pt(0)=1deg 5x(0)=14m8y(0)=52(0)=0 dx(0)=1mdy(0)=dz(0)=0 Controlled Retrieval




0 1 2 Time (Orbits)


CO CD

cdT

CO CD

T J CM

CO

1 2 Time (Orbits)



150

100

1 2 3 0 1 2 Time (Orbits) Time (Orbits)


CO

(a)

x 50

2 3 Time (Orbits)

Figure 5 -5 Retrieval dynamics of the three-body STSS using the non-linear rigid FLT controller with offset along the local vertical and orbit normal (a) attitude and vibration response

108

STSS configuration 3-Body a1(0)=ct2(0)=at(0)=2deg p1(0)=p2(0)=(3(0)=1deg 5x(0)=14m5y(0)=82(0)=0 dx(0)=1mdy(0)=dz(0)=0 Controlled Retrieval



E

1 2 Time (Orbits)


2E-5

1 2 Time (Orbits)


OEOft

1 2 3 Time (Orbits)


1 2 3 Time (Orbits)


1 2 Time (Orbits)


1E-5 E z

2

0E0


Figure 5-5 Retrieval dynamics of the three-body STSS using the non-linear rigid FLT controller with offset along the local vertical and orbit normal (b) control actuator time histories

109

52 Control of Tethers Elastic Vibrations

521 Prel iminary remarks

The requirement of a precisely controlled micro-gravity environment as well

as the accurate release of satellites into their final orbit demands additional control

of the tethers vibratory motion in addition to its attitude regulation To that end

an active vibration suppression strategy is designed and implemented in this section

The strategy adopted is based on offset control ie time dependent variation

of the tethers attachment point at the platform (satellite 1) A l l the three degrees

of freedom of the offset motion are used to control both the longitudinal as well as

the inplane and out-of-plane transverse modes of vibration In practice the offset

controller can be implemented through the motion of a dedicated manipulator or a

robotic arm supporting the tether which in turn is supported by the platform The

focus here is on the control of elastic deformations during stationkeeping (ie fully

deployed tether fixed length) as it represents the phase when the mission objectives

are carried out

This section begins with the linearization of the systems equations of motion

for the reduced three body stationkeeping case This is followed by the design of

the optimal control algorithm Linear Quadratic Gaussian-Loop Transfer Recovery

( L Q G L T R ) based on the reduced model and its implementation on the full nonlinear

model Final ly the systems response in the presence of offset control is presented

which tends to substantiate its effectiveness

522 System linearization and state-space realization

The design of the offset controller begins with the linearization of the equations

of motion about the systems equilibrium position Linearization of extremely lengthy

(even in matrix form) highly nonlinear nonautonomous and coupled equations of

110

motion presents a challenging problem This is further complicated by the fact the

pitch angle ot is not referred to the L V L H frame Hence the equilibrium pitch angle

is not a constant but is equal to the instantaneous orbital angle 9(t) Two methods

are available to resolve this problem

One may use the non-stationary equations of motion in their present form and

derive a controller based on the Linear T ime Varying ( L T V ) system Alternatively

one can design a Linear T ime Invariant (LTI) controller based on a new set of reduced

governing equations representing the motion of the three body tethered system A l shy

though several studies pertaining to L T V systems have been reported[5660] the latter

approach is chosen because of its simplicity Furthermore the L T I controller design

is carried out completely off-line and thus the procedure is computationally more

efficient

The reduced model is derived using the Lagrangian approach with the genershy

alized coordinates given by

where lty and are the pitch and roll angles of link i relative to the L V L H frame

respectively 5X 8y and 5Z are the generalized coordinates associated with the flexible

tether deformations in the longitudinal inplane and out-of-plane transverse direcshy

tions respectively Only the first mode of vibration is considered in the analysis

The nonlinear nonautonomous and coupled equations of motion for the tethshy

ered system can now be written as

Qred = [ltXlPlltX2P2fixSy5za302gt] (514)

M redQred + fr mdash 0gt (515)

where Mred and frec[ are the reduced mass matrix and forcing term of the system

111

respectively They are functions of qred and qred in addition to the time varying offset

position d2 and its time derivatives d2 and d2 A detailed derivation of Eq(515)

is given in Appendix II A n additional consequence of referring the pitch motion to

a local frame is that the new reduced equations are now independent of the orbital

angle 9 and under the further assumption of the system negotiating a circular orbit

91 remains constant

The nonlinear reduced equations can now be linearized about their static equishy

l ibr ium position For all the generalized coordinates the equilibrium position is zero

wi th the exception of 5X which has a non-zero equilibrium 5XQ The offset position

d2) is linearized about its ini t ial position d2^ Linearizing Eq(515) and recasting

into matr ix form gives

Msqred + CsQred + KsQred + Md^2 + Cdd2 + Kdd2 + fs = 0 (516)

where Ms Cs Ks Md Cd Kd and s are constant matrices Mul t ip ly ing throughout

by Ms

1 gives

qr = -Ms 1Csqr - Ms

lKsqr

- Ms~lCdd22 - Ms-Kdd2 - Ms~Mdd2 - Mg~1 fs r-1 (517)

Defining ud = d2 and v = qTd^T Eq(517) can be rewritten as

Pu t t ing

v =

+

-M~lCs -M~Cd -1

0

-Mg~lMd

Id

0 v + -M~lKs -M~lKd

0 0

-M-lfs

0

MCv + MKv + Mlud +Fs

- 4 ) - ( gt the L T I equations of motion can be recast into the state-space form as

(518)

(519)

x =Ax + Bud + Fd (520)

112

where

A = MC MK Id12 o

24x24

and

Let

B = ^ U 5 R 2 4 X 3 -

Fd = I FJ ) e f t 2 4 1

mdash _ i _ -5

(521)

(522)

(523)

(524)

where x is the perturbation vector from the constant equilibrium state vector xeq

Substituting Eq(524) into Eq(520) gives the linear perturbation state equation

modelling the reduced tethered system as

bullJ x mdash x mdash Ax + Bud + (Fd + Axeq) (525)

It can be shown that for ideal control[60] Fd + Axeq = 0 thus giving the familiar

state-space equation

S=Ax + Bud (526)

The selection of the output vector completes the state space realization of

the system The output vector consists of the longitudinal deformation from the

equil ibrium position SXQ of the tether at xt = h the slope of the tether due to the

transverse deformation at xt = 0 and the offset position d2 from its in i t ia l position

d2Q Thus the output vector y is given by

K(o)sx

V

dx ~ dXQ dy ~ dyQ

d z - d z0 J

Jy

dx da XQ vo

d z - dZQ ) dy dyQ

(527)

113

where lt ( 0 ) = ddxi[$vxi)]Xi=o and lt ( 0 ) = ddxi[4gtw(xi)]Xi=0

523 Linear Quadratic Gaussian control with Loop Transfer Recovery

W i t h the linear state space model defined (Eqs526527) the design of the

controller can commence The algorithm chosen is the Linear Quadratic Gaussian

( L Q G ) estimator based optimal controller[6061] The L Q G is a widely used optimal

controller with its theoretical background well developped by many authors over the

last 25 years It involves of the design of the Kalman-Bucy Fi l ter ( K B F ) which

provides an estimate of the states x and a Linear Quadratic Regulator ( L Q R ) which

is separately designed assuming all the states x are known Both the L Q R and K B F

designs independently have good robustness properties ie retain good performance

when disturbances due to model uncertainty are included However the combined

L Q R and K B F designs ie the L Q G design may have poor stability margins in the

presence of model uncertainties This l imitat ion has led to the development of an L Q G

design procedure that improves the performance of the compensator by recovering

the full state feedback robustness properties at the plant input or output (Figure 5-6)

This procedure is known as the Linear Quadratic Guassian-Loop Transfer Recovery

( L Q G L T R ) control algorithm A detailed development of its theory is given in

Ref[61] and hence is not repeated here for conciseness

The design of the L Q G L T R controller involves the repeated solution of comshy

plicated matrix equations too tedious to be executed by hand Fortunenately the enshy

tire algorithm is available in the Robust Control Toolbox[62] of the popular software

package M A T L A B The input matrices required for the function are the following

(i) state space matrices ABC and D (D = 0 for this system)

(ii) state and measurement noise covariance matrices E and 0 respectively

(iii) state and input weighting matrix Q and R respectively

114

u

MODEL UNCERTAINTY

SYSTEM PLANT

y

u LQR CONTROLLER

X KBF ESTIMATOR

u

y

LQG COMPENSATOR

PLANT

X = f v ( X t )

COMPENSATOR

x = A k x - B k y

ud

= C k f

y

Figure 5-6 Block diagram for the LQGLTR estimator based compensator

115

A s mentionned earlier the main objective of this offset controller is to regu-mdash

late the tether vibration described by the 5 equations However from the response

of the uncontrolled system presented in Chapter 4 it is clear that there is a large

difference between the magnitude of the librational and vibrational frequencies This

separation of frequencies allow for the separate design of the vibration and attitude mdash mdash

controllers Thus only the flexible subsystem composed of the S and d equations is

required in the offset controller design Similarly there is also a wide separation of

frequencies between the longitudinal and transverse modes of vibrations permitt ing mdash

the decoupling of the flexible subsystem into a set of longitudinal (Sx and dx) and

transverse (5y Sz dy and dz) subsystems The appended offset system d must also

be included since it acts as the control actuator However it is important to note

that the inplane and out-of-plane transverse modes can not be decoupled because

their oscillation frequencies are of the same order The two flexible subsystems are

summarized below

(i) Longitudinal Vibra t ion Subsystem

This is defined by u mdash Auxu + Buudu

(528)

where xu = SxdX25xdX2T u d u = dX2 and from Eq(527)

0 0 $u(lt) 0 0 0 0 1

0 0 1 0 0 0 0 1

(529)

Au and Bu are the rows and columns of A and B respectively and correspond to the

components of xu

The L Q R state weighting matrix Qu is taken as

Qu =

bull1 0 0 o -0 1 0 0 0 0 1 0

0 0 0 10

(530)

116

and the input weighting matrix Ru = 1^ The state noise covariance matr ix is

selected as 1 0 0 0 0 1 0 0 0 0 4 0

LO 0 0 1

while the measurement noise covariance matrix is taken as

(531)

i o 0 15

(532)

Given the above mentionned matrices the design of the L Q G L T R compenshy

sator is computed using M A T L A B with the following 2 commands

(i) kfu = l q r c ( ^ ( i C bdquo d i a g m x ( E u e u ) )

(ii) [AkugtBkugtCkugtDku =^ryAuBuCuDukfuQuRur)

where r is a scaler

The first command is used to compute the Ka lman filter gain matrix kfu Once

kfu is known the second command is invoked returning the state space representation

of the L Q G L T R dynamic compensator to

(533) xu mdash Akuxu Bkuyu

where xu is the state estimate vector oixu The function l t ry performs loop transfer

recovery at the system output ie the return ratio at the output approaches that

of the K B F loop given by Cu(sld - Au)Kfu[61 This is achieved by choosing a

sufficently large scaler value r in l t ry such that the singular values of the return

ratio approach those of the target design For the longitudinal controller design

r mdash 5 x 10 5 Figure 5-7(a) compares the singular values of the recovered compensator

design and the non-recovered L Q G compensator design with respect to the target

Unfortunetaly perfect recovery is not possible especially at higher frequencies

117

100

co

gt CO

-50

-100 10

Longitudinal Compensator Singular Values i I I I N I I mdash i mdash i i i i n i i mdash i mdash i i i i inimdash imdashi i i i i i i i mdash i mdash i i M i I I

Target r = 5x10 5

r = 0 (LQG)

i i i i I I i ii i i i 1 1 1 1 i i i i i 1 1 1 1 ii i I I i i i i 11111 -3 10 -2

(a)

101 10deg Frequency (rads)

101 10

Transverse Compensator Singular Values

100

50

X3

gt 0

CO

-50

bull100 10

-imdashi i 111II i 111 ni 1mdashi I I I I I I I 1mdashi I I I N I I 1mdashi i M 111

Target r = 50 r = 0 (LQG)

m i i i i i i n i i i i I I I I I I I 1mdashi I I I I I I I 1mdashi i 111 n -3 10 -2

(b)


101 10

Figure 5-7 Singular values for the LQG and LQGLTR compensator compared to target return ratio (a) longitudinal design (b) transverse design

118

because the system is non-minimal ie it has transmission zeros with positive real

parts[63]

(ii) Transverse Vibra t ion Subsystem

Here Xy mdash AyXy + ByU(lv

Dv = Cvxv

with xv = 6y 8Z dV2 dZ2 Sy 6Z dV2 dZ2T] udv = dy2dZ2T and

(534)

Cy

0 0 0 0 (0) 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0

h 0 0 0

0 0 0 lt ( 0 ) o 0

0 1 0 0 0 1

0 0 0

2 r

h 0 0

0 0 1 0 0 1

(535)

Av and Bv are the rows and columns of the A and B corresponding to the components

O f Xy

The state L Q R weighting matrix Qv is given as

Qv

10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

(536)

mdash

Ry =w The state noise covariance matrix is taken

1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 x 10 4 0

0 0 0 0 0 0 0 9 x 10 4

(537)

119

while the measurement noise covariance matrix is

1 0 0 0 0 1 0 0 0 0 5 x 10 4 0

LO 0 0 5 x 10 4

(538)

As in the development of the longitudinal controller the transverse offset conshy

troller can be designed using the same two commands (lqrc and l try) wi th the input

matrices corresponding to the transverse subsystem with r = 50 The result is the

transverse compensator system given by

(539)

where xv is the state estimates of xv The singular values of the target transfer

function is compared with those achieved by the L Q G and L Q G L T R in Figure 5-

7(b) It is apparent that the recovery is not as good as that for the longitudinal case

again due to the non-minimal system Moreover the low value of r indicates that

only a lit t le recovery in the transverse subsystem is possible This suggests that the

robustness of the L Q G design is almost the maximum that can be achieved under

these conditions

Denoting f y = f j f j r xf = x^x^T and ud = udu ^ T the longishy

tudinal compensator can be combined to give

xf =

$d =

Aku

0

0 Akv

Cku o

Cu 0

Bku

0 B Xf

Xf = Ckxf

0

kv

Hf = Akxf - Bkyf

(540)

Vu Vv 0 Cv

Xf = CfXf

Defining a permutation matrix Pf such that Xj = PfX where X is the state vector

of the original nonlinear system given in Eq(32) the compensator and full nonlinear

120

system equations can be combined as

X

(541)

Ckxf

A block diagram representation of Eq(541) is shown in Figure 5-6

524 Simulation results

The dynamical response for the stationkeeping STSS with a tether length of 20

km and ini t ia l offset position of 1 m in each direction is presented in Figure 5-8 when

both the F L T attitude controller and the L Q G L T R offset controller are activated As

expected the offset controller is successful in quickly damping the elastic vibrations

in the longitudinal and transverse direction (Figure 5-8b) However from Figure 5-

8(a) it is clear that the presence of offset control requires a larger control moment

to regulate the attitude of the platform This is due to the additional torque created

by the larger moment arm around 2 m and 125 m in the inplane and out-of-plane

directions respectively introduced by the offset control In addition the control

moment for the platform is modulated by the tethers transverse vibration through

offset coupling However this does not significantly affect the l ibrational motion of

the tether whose thruster force remains relatively unchanged from the uncontrolled

vibration case

Final ly it can be concluded that the tether elastic vibration suppression

though offset control in conjunction with thruster and momentum-wheel attitude

control presents a viable strategy for regulating the dynamics tethered satellite sysshy

tems

121

STSS configuration 3-Body a1(0)=a2(0)=at(0)=2deg P1(0)=p2(0)=pt(0)=1deg 8x(0)=14m6y(0)=52(0)=1m dx(0)=dy(0)=dz(0)=1m Stationkeeping I = 20km LQGLTR Vibration Control

Sat 1 Pitch and Roll Angles Tether Pitch and Roll Angles mdash 1 1 1 1 1 1 I 1 1 lt 1 ~ ^ 1 ^ l _

Time (Orbits) Time (Orbits) Sat 1 Roll Control Moment Tether Roll Control Thrust


Figure 5-8 Controlled dynamics of the three-body STSS during stationkeeping using the nonlinear rigid FLT attitude controller and LQGLTR offset vibration controller (a) attitude and libration controller reshysponse

122

STSS configuration 3-Body a1(0)=a2(0)=a(0)=2deg p1(0)=(32(0)=Q(0)=1deg ox(0)=14mSy(0)=5z(0)=1m dx(0)=dy(0)=dz(0)=1m Stationkeeping lt = 20km LQGLTR Vibration Control

Tether X Offset Position

Time (Orbits) Tether Y Tranverse Vibration 20r

Time (Orbits) Tether Y Offset Position

t o

1 2 Time (Orbits)

Tether Z Tranverse Vibration Time (Orbits)

Tether Z Offset Position

(b)

Figure 5-8


Controlled dynamics of the three-body STSS during stationkeeping using the nonlinear rigid FLT attitude controller and LQGLTR offset vibration controller (b) vibration and offset response

123

6 C O N C L U D I N G R E M A R K S

61 Summary of Results

The thesis has developed a rather general dynamics formulation for a multi-

body tethered system undergoing three-dimensional motion The system is composed

of multiple rigid bodies connected in a chain configuration by long flexible tethers

The tethers which are free to undergo libration as well as elastic vibrations in three

dimensions are also capable of deployment retrieval and constant length stationkeepshy

ing modes of operation Two types of actuators are located on the rigid satellites

thrusters and momentum-wheels

The governing equations of motion are developed using a new Order(N) apshy

proach that factorizes the mass matrix of the system such that it can be inverted

efficiently The derivation of the differential equations is generalized to account for

an arbitrary number of rigid bodies The equations were then coded in FORTRAN

for their numerical integration with the aid of a symbolic manipulation package that

algebraically evaluated the integrals involving the modes shapes functions used to dis-

cretize the flexible tether motion The simulation program was then used to assess the

uncontrolled dynamical behaviour of the system under the influence of several system

parameters including offset at the tether attachment point stationkeeping deployshy

ment and retrieval of the tether The study covered the three-body and five-body

geometries recently flown OEDIPUS system and the proposed BICEPS configurashy

tion It represents innovation at every phase a general three dimensional formulation

for multibody tethered systems an order-N algorithm for efficient computation and

application to systems of contemporary interest

Two types of controllers one for the attitude motion and the other for the

124

flexible vibratory motion are developed using the thrusters and momentum-wheels for

the former and the variable offset position for the latter These controllers are used to

regulate the motion of the system under various disturbances The attitude controller

is developed using the nonlinear Feedback Linearization Technique ( F L T ) and is based

on the nonlinear but rigid model of the system O n the other hand the separation

of the longitudinal and transverse frequencies from those of the attitude response

allows for the development of a linear optimal offset controller using the robust Linear

Quadratic Gaussian-Loop Transfer Recovery ( L Q G L T R ) method The effectiveness

of the two controllers is assessed through their subsequent application to the original

nonlinear flexible model

More important original contributions of the thesis which have not been reshy

ported in the literature include the following

(i) the model accounts for the motion of a multibody chain-type system undershy

going librational and in the case of tethers elastic vibrational motion in all

the three directions

(ii) the dynamics formulation is based on a recursive O(N) Lagrangian algorithm

that efficiently computes the systems generalized acceleration vector

(iii) the development of an attitude controller based on the Feedback Linearizashy

tion Technique for a multibody system using thrusters and momentum-wheels

located on each rigid body

(iv) the design of a three degree of freedom offset controller to regulate elastic v i shy

brations of the tether using the Linear Quadratic Gaussian and Loop Transfer

Recovery method ( L Q G L T R )

(v) substantiation of the formulation and control strategies through the applicashy

tion to a wide variety of systems thus demonstrating its versatility

125

The emphasis throughout has been on the development of a methodology to

study a large class of tethered systems efficiently It was not intended to compile

an extensive amount of data concerning the dynamical behaviour through a planned

variation of system parameters O f course a designer can easily employ the user-

friendly program to acquire such information Rather the objective was to establish

trends based on the parameters which are likely to have more significant effect on

the system dynamics both uncontrolled as well as controlled Based on the results

obtained the following general remarks can be made

(i) The presence of the platform products of inertia modulates the attitude reshy

sponse of the platform and gives rise to non-zero equilibrium pitch and roll

angles

(ii) Offset of the tether attachment point significantly affects the equil ibrium orishy

entation of the system as well as its dynamics through coupling W i t h a relshy

atively small subsatellite the tether dynamics dominate the system response

with high frequency elastic vibrations modulating the librational motion On

the other hand the effect of the platform dynamics on the tether response is

negligible As can be expected the platform dynamics is significantly affected

by the offset along the local horizontal and the orbit normal

(iii) For a three-body system deployment of the tether can destabilize the platform

in pitch as in the case of nonzero offset However the roll motion remains

undisturbed by the Coriolis force Moreover deployment can also render the

tether slack if it proceeds beyond a critical speed

(iv) Uncontrolled retrieval of the tether is always unstable in pitch It also leads

to high frequency elastic vibrations in the tether

(v) The five-body tethered system exhibits dynamical characteristics similar to

those observed for the three-body case As can be expected now there are

additional coupling effects due to two extra bodies a rigid subsatellite and a

126

flexible tether

(vi) Cartwheeling motion of the B I C E P S configuration can also be used to adshy

vantage in the deployment of the tether However exceeding a crit ical ini t ia l

cartwheeling rate can result in a large tension in the tether which eventually

causes the satellite to bounce back rendering the tether slack

(vii) Spinning the tether about its nominal length as in the case of O E D I P U S

introduces high frequency transverse vibrations in the tether which in turn

affect the dynamics of the satellites

(viii) The F L T based controller is quite successful in regulating the attitude moshy

tion of both the tether and rigid satellites in a short period of time during

stationkeeping deployment as well as retrieval phases

(ix) The offset controller is successful in suppressing the elastic vibrations of the

tether wi thin the allowable l imits of the offset mechanism ( plusmn 2 0 m)

62 Recommendations for Future Study

A s can be expected any scientific inquiry is merely a prelude to further efforts

and insight Several possible avenues exist for further investigation in this field A

few related to the study presented here are indicated below

(i) inclusion of environmental effects particularly the free molecular reaction

forces as well as interaction with Earths magnetic field and solar radiation

(ii) addition of flexible booms attached to the rigid satellites providing a mechashy

nism for energy dissipation

(iii) validation of the new three dimensional offset control strategies using ground

based experiments

(iv) animation of the simulation results to allow visual interpretation of the system

dynamics

127

BIBLIOGRAPHY

[1] Tsiolkovsky K E Speculation Between Ear th and Sky izd-vo A N - S S S R Scishyence Fiction Works 1859 (reprinted 1959)

[2] Starly W H and Adlhock R W Study of the Retrieval of an Astronaut from an Extra-Vehicular Assignment TMC Report No S-356 November 1963

[3] Lang D L and Nolt ing R K Operations with Tethered Space Vehicles NASA SP-138 Gemini Summary Conference February 1967

[4] Sasaki S et al Results from a Series of Tethered Rocket Experiments Jourshynal of Spacecraft and Rocket V o l 24 No 5 1987 pp444-453

[5] Gullahorn G E et al Observations of Tethered Satellite System (TSS-1) Dynamics AASAIAA Astrodynamics Specialist Conference Vic tor ia Canada August 1993 Paper No A A S 93-704

[6] Tyc G et al Dynamics and Stability of a Spinning Tethered Spacecraft wi th Flexible Appendages Advances in Astronautical Sciences Editors A r u n K Misra et al V o l 85 Part I 1993 pp 877-896

[7] Bor tolami S B et al Control Law for the Deployemnt of S E D S - I I AASAIAA Astrodynamics Specialist Conference Vic tor ia Canada 1993 Paper No A A S 93-706

[8] James H G BIstatic Canadian Experiment on Plasmas in Space ( B I C E P S ) CASI Symposium on Small Satellites Ottawa Canada November 1993 pp 36-45

[9] Rupp C C A Tether Tension Control Law for Tethered Sub-satellite Deployed along Local Vert ical NASA TM X-64963 September 1975

[10] Baker P W et a l Tethered Subsatellite Study NASA TM X-73314 March 1976

[11] M o d i V J and Misra A K Deployment Dynamics of a Tethered Satellite System AIAAAAS Astrodynamics Conference Palo A l t o California August 1978 Paper No 78-1398

[12] Mis ra A K and M o d i V J Deployment and Retrieval of Shuttle Supported Tethered Satellite Journal of Guidance Control and Dynamics V o l 5 No 3 1982 pp 278-285

[13] Misra A K and M o d i V J A Survey on the Dynamics and Control of Tethshyered Satellite Systems NASAAl A APSN International Conference on Tethers Arl ington V A U S A September 1986 Paper No AAS-86-246 also Advances in the Astronautical Sciences Editors P M Bainum et al American Astro-nautical Society V o l 62 pp 667-719

[14] Mis ra A K Amier Z and M o d i V J Atti tude Dynamics of Three-Body Tethered Systems Acta Astronautica V o l 17 No 10 1988 pp 1059-1068

[15] Mis ra A K and M o d i V J Three-Dimensional Dynamics and Control of

128

Tether-Connected N - B o d y Systems Acta Astronautica V o l 26 No 2 1992 pp 77-84

[16] Keshmir i M and Misra A K A Genaral Formulation for N - B o d y Tethered Satellite System Dynamics Advances in Astronautical Sciences Editors A r u n K Misra et al V o l 85 Part I 1993 pp 623-644

[17] Rosenthal D E Triangularization of Equations of Mot ion for Robot ic Sysshytems Journal of Guidance Control and Dynamics V o l 11 No 3 May-June 1988 pp 278-281

[18] Book W J Recursive Lagrangian Dynamics of Flexible Manipulator Arms The International Journal of Robotics Research V o l 3 No 3 1984 pp 87-101

[19] Fu K S Gonzalez R C and Lee C S G Robotics Control Sensing Vis ion and Intelligence M c G r a w - H i l l Publishing Company New York U S A 1987 pp 103-124

[20] van Woerkom P T h L M and de Boer A Development and Val idat ion of a Linear Recursive Order-N Algor i thm for the Simulation of Flexible Space Manipulator Dynamics Acta Astronautica V o l 35 No 23 1995 pp 175-185

[21] Hollerbach J M A Recursive Lagrangian Formulation of Manipulator Dyshynamics and a Comparative Study of Dynamics Formulation Complexity IEEE Transactions on Systems Man and Cybernetics V o l SMC-10 No 11 Novemshyber 1980 pp 730-736

[22] Bae D S and Haug E J A Recursive Formulation for Constrained Meshychanical System Dynamics Part I Open Systems Mechanical Structures and Machines V o l 15 No 3 1987 pp 359-382

[23] Rosenthal D E A n Order n Formulation for Robotic Systems The Journal of the Astronautical Sciences V o l 38 No 4 October-December 1990 pp 511-529

[24] Keat J E Mul t ibody System Order n Dynamics Formulation Based on Velocshyity Transform Method Journal of Guidance Control and Dynamics V o l 13 No 2 March -Apr i l 1990 pp 207-212

[25] Kurd i l a A J Menon R G and Sunkel J W Nonrecursive Order N Formushylation of Mul t ibody Dynamics Journal of Guidance Control and Dynamics V o l 16 No 5 September-October 1993 pp 838-844

[26] Ja in A Unified Formulation of Dynamics for Serial R ig id Mul t i body Systems Journal of Guidance Control and Dynamics V o l 14 No 3 May-June 1991 pp 531-542

[27] Rodriguez G and Kreutz-Delgato K Spatial Operator Factorization and Inshyversion of the Manipulator Mass Mat r ix IEEE Transaction on Robotics and Automation V o l 8 No 1 February 1992 pp 65-76

[28] Ja in A and Rodriguez G Recursive Flexible Mul t ibody Systems Dynamics Using Spatial Operators Journal of Guidance Control and Dynamics V o l 15 No 6 November-December 1992 pp 1453-1466

129

[29] Jain A and Rodriguez G Recursive Dynamics Algor i thm for Mul t ibody Sysshytems with Prescribed Mot ion Journal of Guidance Control and Dynamics V o l 16 No 5 September-October 1993 pp 830-837

[30] Pradhan S M o d i V J and Misra A K A n Order N Dynamics Formulation of Flexible Mul t ibody Systems in Tree Topology - The Lagrangian Approach AIAAAAS Astrodynamics Specialist Conference San Diego C A U S A July 1996 Paper No 96-3624 pp 480-490

[31] Banerjee A K Order-n Formulation of Extrusion of a Beam with Large Bendshying and Rotation Journal of Guidance Control and Dynamics V o l 15 No 1 January-February 1992 pp 121-127

[32] X u D M The Dynamics and Control of the Shuttle Supported Tethered Sub-satellite System P h D Thesis Department of Mechanical Engineering M c G i l l University November 1984

[33] X u D M Misra A K and M o d i V J On Vibra t ion Control of Tethered Satellite Systems Proceedings of theNASAJPL Workshop on Applications of Distributed Systems Theory to the Control of Large Space Structures Pasadena California July 1982 pp 317-327

[34] Pradhan S M o d i V J and Misra A K On the Inverse Control of the Tethered Satellite Systems The Journal of the Astronautical Sciences A A S Publications Office San Diego California U S A V o l 43 No 2 Apr i l - June 1995 pp 179-193

[35] K i m E and Vadal i S R Modeling Issues Related to Retrieval of Flexible Tethered Satellite Systems Journal of Guidance Control and Dynamics V o l 18 No 5 September-October 1995 pp 1169-1176

[36] W u S Chang C and Housner J M Finite Element Approach for Transient Analysis of Mul t ibody Systems Journal of Guidance Control and Dynamics V o l 15 No 4 July-August 1992 pp 847-854

[37] Pradhan S M o d i V J and Misra A K Simultaneous Control of Platform Att i tude and Tether Vibra t ion using Offset Strategy AIAAA AS Astrodynamics Specialist Conference San Diego C A U S A July 1996 Paper No 96-3570 pp 1-11

[38] Ruying F and Bainum P M Dynamics and Control of a Space Platform with a Tethered Subsatellite Journal of Guidance Control and Dynamics V o l 11 No 4 July-August 1988 pp 377-381

[39] M o d i V J Lakshmanan P K and Misra A K Offset Control of Tethered Satellite Systems Analysis and Experimental Verification Acta Astronautica V o l 21 No 5 1990 pp 283-294

[40] X u D M Misra A K and M o d i V J Thruster Augmented Act ive Cin t ro l of a Tethered Satellite System During its Retrieval Journal of Guidance Control and Dynamics V o l 9 No 6 November-December 1986 pp 663-672

[41] M o d i V J Pradhan S and Misra A K Off-Set Control of the Tethered Systems Using a Graph Theoretic Approach Acta Astronautica V o l 35 No

130

6 1995 pp 373-384

[42] Pradhan S M o d i V J Bhat M S and Misra A K Mat r ix Method for Eigenstructure Assignment The Mult i - Input Case with Appl ica t ion Journal of Guidance Control and Dynamics V o l 17 No 5 September-October 1994 pp 983-989

[43] Pradhan S Planar Dynamics and Control of Tethered Satellite Systems P h D Thesis Department of Mechanical Engineering The University of Br i t i sh Columbia December 1994

[44] Bainum P M and Kumar V K Optimal-Control of the Shuttle-Tethered System Acta Astronautica V o l 7 No 12 1980 pp 1333-1348

[46] Hughes P C Spacecraft Att i tude Dynamics John Wi ley amp Sons New York U S A 1992 Chapter 2 pp 7-38

[47] Stengel R F Optimal Control and Estimation Dover Publications Inc New York U S A 1994 p 26

[48] Nayfeh A H and Mook D T Nonlinear Oscillation John Wi ley and Sons New York U S A 1979 pp 485-500

[49] He X and Powell J D Tether Damping in Space Journal of Guidance

Control and Dynamics V o l 13 No 1 1990 pp 104-112

[50] Meirovitch L Elements of Vibration Analysis M c G r a w - H i l l Inc New York U S A 1986 pp 255-256

[51] Greenwood D T Principles of Dynamics Prentice Ha l l Inc Englewood Cliffs New Jersey U S A 1965 pp 252-275

[52] Boyce W E and D i P r i m a R C Elemental Differential Equations and Boundshy

ary Value Problems John Wi ley amp Sons Inc New York U S A 1986 pp 424-431

[53] IMSL Library Reference Manual Vol 1 I M S L Inc Houston Texas U S A 1980 pp D G E A R 1 - D G E A R 9

[54] Gear C W Numerical Initial Value Problems in Ordinary Differential Equashy

tions Prentice-Hall Englewood Cliffs New Jersey U S A 1971 pp 158-166

[55] Char B W Geddes K O Gonnet G H Leong B L Monagan M B and Wat t S M First Leaves A Tutorial Introduction to Maple V Springer-Verlag New York U S A 1992

[56] Tsakalis K S and Ioannou P A Linear Time-Varying Systems Control and

Adaptation Prentice-Hall Inc Englewood Cliffs New Jersey U S A 1993 pp 148-180

[57] Su R O n the Linear Equivalents of Nonlinear Systems Systems and Control

Letter V o l 2 No 1 1982 pp 48-52

[58] M o d i V J Karray F and Chan J K On the Control of a Class of Flexible Manipulators Using Feedback Linearization Approach 42nd Congress of the International Astronautical Federation October 1991 Montreal Canada Paper No IAF-91-324

131

[59] K u o B C Automatic Control Systems Prentice-Hall Inc Englewood Cliffs New Jersey U S A 1987 pp 314-326

[60] Athans M The Role and Use of the Stochastic Linear-Quadrat ic-Guassian Problem in Control System Design IEEE Transactions on Automatic Control V o l A C - 1 6 No 6 December 1971 pp 529-552

[61] Maciejowski J M Multivariable Feedback Design Addison-Wesley Publ ishing Company Wokingham England 1989 Chapters 1-5

[62] Chiang R Y and Safonov M G Robust Control Toolbox Users Guide The M a t h Works Inc Natick Mass U S A 1992 pp 272-274

[63] Stein G and Athans M The L Q G L T R Procedure for Mult ivariable Feedback Control Design IEEE Transactions on Automatic Control V o l A C - 3 2 No 2 February 1987 pp 105-114

[64] Borisenko A I and Tarapov I E Vector and Tensor Analysis with Applicashytions Dover Publications Inc New York U S A 1979

[65] Lass H Vector and Tensor Analysis M c G r a w - H i l l Book Company Inc New York U S A 1950

132

A P P E N D I X I TENSOR R E P R E S E N T A T I O N OF T H E

EQUATIONS OF M O T I O N


The derivation of the equations of motion for the multibody tethered system

involves the product of several matrices as well as the derivative of matrices and

vectors with respect to other vectors In order to concisely code these relationships

in FORTRAN the matrix equations of motion are expressed in tensor notation

Tensor mathematics in general is an extremely powerful tool which can be

used to solve many complex problems in physics and engineering[6465] However

only a few preliminary results of Cartesian tensor analysis are required here They

are summarized below

12 Mathematical Background

The representation of matrices and vectors in tensor notation simply involves

the use of indices referring to the specific elements of the matrix entity For example

v = vk (k = lN) (Il)

A = Aij (i = lNj = lM)

where vk is the kth element of vector v and Aj is the element on the t h row and

3th column of matrix A It is clear from Eq(Il) that exactly one index is required to

completely define an entire vector whereas two independent indices are required for

matrices For this reason vectors and matrices are known as first-order and second-

order tensors respectively Scaler variables are known as zeroth-order tensors since

in this case an index is not required

133

Matrix operations are expressed in tensor notation similar to the way they are

programmed using a computer language such as FORTRAN or C They are expressed

in a summation notation known as Einstein notation For example the product of a

matrix with a vector is given as

w = Av (12)

or in tensor notation W i = ^2^2Aij(vkSjk)

4-1 ltL3gt j

where 8jk is the Kronecker delta Dropping the summation symbol the matrix-vector

product can be expressed compactly as

Wi = AijVj = Aimvm (14)

Here j is a dummy index since it appears twice in a single term and hence can

be replaced by any other index Note that since the resulting product is a vector

only one index is required namely i that appears exactly once on both sides of the

equation Similarly

E = aAB + bCD (15)

or Ej mdash aAimBmj + bCimDmj

~ aBmjAim I bDmjCim (16)

where A B C D and E are second-order tensors (matrices) and a and b are zeroth-

order tensors (scalers) In addition once an expression is in tensor form it can be

treated similar to a scaler and the terms can be rearranged

The transpose of a matrix is also easily described using tensors One simply

switches the position of the indices For example let w = ATv then in tensor-

notation

Wi = AjiVj (17)

134

The real power of tensor notation is in its unambiguous expression of terms

containing partial derivatives of scalers vectors or matrices with respect to other

vectors For example the Jacobian matrix of y = f(x) can be easily expressed in

tensor notation as

di-dx-j-1 ( L 8 )

mdash which is a second-order tensor as expected If f(x) = Ax where A is constant then

f - a )

according to the current definition If C = A(x)B(x) then the time derivative of C

is given by

C = Ax)Bx) + A(x)Bx) (110)

or in tensor notation

Cij = AimBmj + AimBmj (111)

= XhiAimfcBmj + AimBmj^k)

which is more clear than its equivalent vector form Note that A^^ and Bmjk are

third-order tensors that can be readily handled in F O R T R A N

13 Forcing Function

The forcing function of the equations of motion as given by Eq(260) is

- u 1-SdM- dqtdPe 9 f i m F(q qt) = Mq - -q mdashq + mdashmdash - Qd (260)

This expression can be converted into tensor form to facilitate its implementation

into F O R T R A N source code

Consider the first term Mq From Eq(289) the mass matrix M is given by

T M mdash Rv MtRv which in tensor form is

Mtj = RZiM^R^j (112)

135

Taking the time derivative of M gives

Mij = qsRv

ni^MtnrnRv

mj + Rv

niMtnTn^Rv

mj + Rv

niMtnmRv

mjs) (113)

The second term on the right hand side in Eq(260) is given by

-JPdMu 1 bdquo T x

2Q ~cWq = 2Qs s r k Q r ^ ^

Expanding Msrk leads to

M src = RnskMtnmRmr + RnsMtnmkRmr + RnsMtnmRmrk- (L15)

Finally the potential energy term in Eq(260) is also expanded into tensor

form to give dPe = dqt dPe

dq dq dqt = dq^QP^

and since ^ = Rp(q)q then

Now inserting Eqs(I13-I17) into Eq(260) and rearranging the tensor form of the

forcing term can how be stated as

Fk(q^q^)=(RP

sk + R P n J q n ) ^ - - Q d k

+ QsQr | (^Rlks ~ 2 M t n m R m r

+ (^RnkMtnms ~ 2~RnsMtnmgtk^ R m r

(^RnkRmrs ~ ~2RnsRmrk^j

where Fk(q q t) represents the kth component of F

(118)

136

A P P E N D I X II R E D U C E D EQUATIONS OF M O T I O N

II 1 Prel iminary Remarks

The reduced model used for the design of the vibration controller is represented

by two rigid end-bodies capable of three-dimensional attitude motion interconnected

with a fixed length tether The flexible tether is discretized using the assumed-

mode method with only the fundamental mode of vibration considered to represent

longitudinal as well as in-plane and out-of-plane transverse deformations In this

model tether structural damping is neglected Finally the system is restricted to a

nominal circular orbit

II 2 Derivation of the Lagrangian Equations of Mot ion

Derivation of the equations of motion for the reduced system follows a similar

procedure to that for the full case presented in Chapter 2 However in this model the

straightforward derivation of the energy expressions for the entire system is undershy

taken ignoring the Order(N) approach discussed previously Furthermore in order

to make the final governing equations stationary the attitude motion is now referred

to the local LVLH frame This is accomplished by simply defining the following new

attitude angles

ci = on + 6

Pi = Pi (HI)

7t = 0

where and are the pitch and roll angles respectively Note that spin is not

considered in the reduced model hence 7J = 0 Substituting Eq(IIl) into Eq(214)

137

gives the new expression for the rotation matrix as

T- CpiSa^ Cai+61 S 0 S a + e i

-S 0 c Pi

(II2)

W i t h the new rotation matrix defined the inertial velocity of the elemental

mass drrii o n the ^ link can now be expressed using Eq(215) as

(215)

Since the tether length is assumed to be fixed T j f j and Tj$jlt^ of Eq(215) are

identically zero Also defining

(II3) Vi = I Pi i

Rdm- f deg r the reduced system is given by

where

and

Rdm =Di + Pi(gi)m + TîSi

Pi(9i)m= [Tai9i T0gi T^] fj

(II4)

(II5)

(II6)

D1 = D D l D S v

52 = 31 + P(d2)m +T[d2

D3 = D2 + l2 + dX2Pii)m + T2i6x

Note that $ j for i = 2 is defined in Section 212 whereas for i = 1 and 3 it is

the null matrix Since only one mode is used for each flexible degree of freedom

5i = [6Xi6yi6Zi]T for i = 2

The kinetic energy of the system can now be written as

i=l Z i = l Jmi RdmRdmdmi (UJ)

138

Expanding Kampi using Eq(II4)

KH = miDD + 2DiP

i j gidmi)fji + 2D T ltMtrade^

+tf J P^(9i)Pi9i)d^l + 2mT j PFmTi^dm^ (II8)

where the integrals are evaluated using the procedure similar to that described in

Chapter 2 The gravitational and strain energy expressions are given by Eq(247)

and Eq(251) respectively using the newly defined rotation matrix T in place of

TV

Substituting the kinetic and potential energy expressions in

d ( 8Ke d(Ke - Pe) i bull i Q ^ 0 (II9)

d t dqred d(ired

with

Qred = [ algt l gt a 2 2 A 2 A 2 lt ^ 2 Q 3 3 ] (5-14)

and d2 regarded as a time-varying parameter the nonlinear non-autonomous equashy

tions of motion for the reduced system can finally be expressed as

MredQred + fred = reg- (5-15)

139

In presenting this thesis in partial fulfilment of the requirements for an advanced

degree at the University of British Columbia I agree that the Library shall make it

freely available for reference and study I further agree that permission for extensive

copying of this thesis for scholarly purposes may be granted by the head of my

department or by his or her representatives It is understood that copying or

publication of this thesis for financial gain shall not be allowed without my written

permission

Department of

The University of British Columbia Vancouver Canada

DE-6 (288)

ABSTRACT























ii















21 Kinematics 14







i i i
























iv























v





A P P E N D I C E S








v i

LIST OF SYMBOLS





respectively








true anomaly




tion








m

Pi

w 0 i

v i i






modulus E






K A Z $i(k + dXi+1)











0N) order-N




matrix

v i i i








th



respectively





in the ith link






F-Q mdash








ix




- $ T



qv---QtNT


adjacent links)



i + $DJi




tions respectively





LIST OF FIGURES


satellite system 2

























xi













without offset










without offset






x i i






















the local vertical








x i i i








xiv



















xv

1 INTRODUCTION























1


2

Satellite N


2


3



























4





tether





















5


lt mdash

lt Satellite


6


























7















ML+F = Q





be considered






8




























9



























10



























11

two methods[39]
























12





motion



modes










( L Q G L T - R )








13


21 Kinematics















frame



Di + Ti(i + f(i)) (21)

14

15


Ui nvi

wi I I i = 1

nzi bull

(22)



taken as 2 j - l

Hi(xiii)=[j) (23)




k (24)





ff(i) = $i(xili)5i(t) (25)


0 0

^i(xiJi) =


0

0 0

0

G 5R 3 x n f

(26)



16






( 3 d

dx~ + dT)^Xili^ ( 2 8 )

one arrives at











1 P i tch F[ = Cf ( a j )F 0



17

where

c f (A) =

-Sai Cai

0 0

and

C(7i ) =

0 -s0i] = 0 1 0

0 cpi

i 0 0 = 0

0

(211)

(212)

(213)








(214) SliCPi CliCPi









18



















(219)

that

(220)

19


20


cos(0i) 0 0 s i n ( 0 i ) 0 0

0 0 1

Dri

h (221)

= DTlDSv




1 1 1 1 J (222)

= D D l D 3 v


(223)



0 0 1

DTl = D D i = I d 3 i = 2N (224)


2N (225)








21





(226)





and

Ai = i(ti) - ^(t 0 -) = - ^ ( t 2 i ) (227)


V = lUli l^lL (228)

1 tf-tQ v y

A t = (229)


221 Kinetic energy




4 ^ ^ PM) T i $ 4 TiSi]i[t (231)

22

where

m

k J

x l (232)



rewritten as

1 T

(233)


Mt =



pound Sfttiqxnq

(234)




i=l (235)



(236) Mt =

0 0

0 Mt2

0

0 0

MH

0 0 0

0 0 0 bull MtN



23


Mti

m D D l


sym 0

0


0 Id nfj

0 m~

i = 1 3 5 N

(237)


gets

D D l

1 D D l = 1 0 0 0 D n

2 0 (238) 0 0 1










sym sym



tr(ffTai

TTaifi) tr(ffTai





(241)



24

and

j tr(A) =tr(^J A^j (243)



P(fi)Pii)dmi =

tr(Tai

TTaiImi) tTTai


TT7Jmi) sym sym sym

(244)


expanded is


sym sym


sym J z^dmi J


(245)






can be written as

V9i f dm f


dm

inn Lgti + T j ^ | (246)





(2-lt 7)

25






readily that



224 Strain energy












(249)




26




by





number of modes







EX^Ei+jEi (252)





experimentally [49]

27


jet = (254) wo-





(ii) Jo




dWd

^ = - i p = 2 4 ( 2 5 7 )

= 0 i = 13 N









28

system

Mq t)t+ F(q q t) = Q(q q t) (259)


















Let

e Unqxl (261) m


29











3 = 1



k=l



Qt Jfe=l

where

and


T fc-1 0 TkKAk

0 0 0 0 0 0 0 0 0 0 0 0 -

r T i - i 0 0 0 0 0 0 0 0 0 G

0 0 0 1

Qt = RpQ

G R n lt x l

(265)

(266)

G Unq x l (267)

(268)

30

where

RP

R 1 0 0 R R 2 0 R RP

2 R 3

0 0 0

R RP R PN _ 1 RN J

(269)




transformations



t h l ink as

D = A - l + Ti_iK + + K A i _ i ^ _ i


(271) (











31




t = Rvil (2-74)


Here

Rv


0 0 0

T3V nN-l

R bullN

e Mnqqxl

Ri =


e K n lt x l

RS

T i-1 0 0 o-0 0 0 0 0 0

0 0 0 1

G Mnq x l

(275)

(276)

(277)




(278)


0 0

0 0

0 0

G $lnq x l


(279)

pound = Rnjt + Rdq


32

where

and

- 0 0 0 RI 0 0

Rn = 0 0

0 0

Rf 0 0 0 rgtd K2 0

Rd = 0 0 Rl

0 0 0

RN-1 0

G 5R n w x l

0 0 0 e 5 R N ^ X L

(281)

(282)






where

D D T i


DTL 0 0 0 d 3 0 0 0 ID

0 0 nfj 0

RP =


2 R3

iJfDTTi i^

x l

0 0 0

R PN-I RN

(284)

(285)



33



0 0 0 0 0 0 0 0

(286)









M = RV MtRv


(289)

(290)

Inverting Eq(290)



34


Mr1

h 0 0

0 0

0 0

0 0 0

0

0

0

tN

(292)



operation








Mq + f = P C A (293)




pcTM-pc

= ( dc + PdegTM-^f) (294)




35


















nu ^5 Qk=E Pern ^ (2-97)

m=l deg Q k






36






Defining


= d + dfY + T 3 f c m 3



(298)

(299)



(2100)

as i - 2

k=l i1 cm (2101)



WjRi A R i (2102)




1 Q

Q3

3 -1

T 3 T t 2 (2103)

37

Satellite 3

mdash gt -raquo cm

D1=d1

mdash gt

a mdash gt

o


38




Q

~Qgt1

0

0

( T ^ 2 ) (2104)







(2105)

where





3

Qk = ^ F e i

i=l

d

dqk (2106)




39


40

be written as


QiMmi



(2107)

(2108)



Q

Q T 3 0 Q T 5 0

0 0 0

0 lt33T3 Qyen Q5

3T5 0

0 0 0 Q4T5 0

0 0 0 Q5

5T5

M M I

Ti 2 m 3

Quu (2109)


41










numerically














42















by letting

thus

X - ^ ^ f ^ ) =ltbullbull) (3 -2 )







43








plott ing package



















44

INIT

DGEAR PARAM

IC amp SYS PARAM

MAIN

lt

DGEAR

lt

FCN

OUTPUT



VIBRATION

r OFFS DYNA

ET MICS


45









longer stored




the system











46

h =

1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760



0 400 0 0 0 400












fcm3 = 10 0T










47




OOEOh

-50E-12

-10E-11 I i i



48



(a)

20E6

10E6r-

00E0

-10E6

-20E6


-

1 I I I 1 I I

I I

AP e ~_

i i 1 i i i i

00E0

UJ -50E-12 U J

lt


-10E-11h

(b) Time (Orbits)


49


























50

a p(0) = 2deg 0(0) = 2C

6(0) = 08 m

0(0) = 001 m


D p y = 0 D p 2 = 0

a oo

c -ooo -0 01


51






^MAAAAAAAWVWWWWVVVW

V bull i bull i I i J 00 05 10 15 20


Time (Orbits)


52
























53


1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760

h =

h = 200 0 0

0 400 0 0 0 400















mdash







54



55




























56







57







(b) Time (Orbits)


58










by the coupling

















59








60



Time (Orbits)



Time (Orbits)



0 1 2 3 4 5 (b) Time (Orbits)


61





Tether Pitch Angle


Tether Roll Angle

CD

edT

1 2 3 Time (Orbits)




CD

C O

(a)

Figure 4-




62



0 1 2 3 4 5







0 1 2 3 4 5 (b) Time (Orbits)


63



2 3 4 Time (Orbits)

Tether Pitch Angle


Tether Roll Angle



2 3 4 Time (Orbits)


(a)

Figure 4-


2 3 4 Time (Orbits)


64




(b) Time (Orbits)


65










V0 = 146 ms
















66



i i i i i i i i i i i 0 1 2 3 4 5 0 1 2 3 4 5






67



0 1 2 3 4 5


(b) Time (Orbits)


68




E





Tether Pitch Angle


Tether Roll Angle

CD

oo ax


_ 1 _ L





-50

(a) 1 2 3 4


Time (Orbits)


69




to

-02

(b)

Figure 4-7

2 3 Time (Orbits)


0 1 1 bull 1 1



5

1 1 1 1 1 i i | i i | ^

_ i i i i L -I



70












roll

45 Tether Retrieval










and subsatellite



71




E


1 2 3 Time (Orbits)




Tether Pitch Angle

1 2 3 Time (Orbits)

Tether Roll Angle

i i i i J I I i _





D )

T J ^ 0 E 0 eg

CQ

-3E-3

(a) 1 2 3 4


Time (Orbits)


7 2





n 1 1 1 f

I I I _j I I I I I 1 1 1 1 1 2 3 4 5



Time (Orbits)


(b)

Figure 4-8

73




(a) Time (Orbits)



i i i i i i i i i i I 0 1 2 3






04- -



74




E 10 to


1 2 Time (Orbits)


(b) Time (Orbits)


75

























76


77



o 1 2 Time (Orbits)


0 1 2 Time (Orbits)






78


p1(0)=p2(0)=(33(0)=pt1(0)=pt2(0)=10





79



























80






i i i i i 0 1 2 3 4 5










81






82



























83



I- I i t i i i i I H 0 1 2 0 1 2





84



laquo = 5 E - 3 |

000 002



Time (Orbits)


85




E

2000

cu T3

2000

co CD

2000

C D CD

T J


1 2 Time (Orbits)


cn CD

33 cdl




1 2 Time (Orbits)




86



5E-3h

co

0E0

025

000 E to

-025

002

-002

(b)

Figure 4-14




Time (Orbits)


87






















88





89


1E-2

to

OEO

E 00

to


005



Time (Orbits)


90





91




92

























93


present case













mance











94





by Eq(2109)

Lett ing

~fr = fr~ QuU (52)


ir = s(frMrdc) (53)



mdash = Fr - Qru





Ire = Frc ~~ Qrcu

(56)

mdash vrci






an(^ qrcd

95









-In ) 0 5 V T ^ r (59)


V (510)








qrcd 37Vxl (511)

96


e U 3 N x l (512)

E K 3 7 V x l (513)

















97

and

o 0

Qrcj mdash

o o

0 0

Oiit) 0 0

1

N

N

1deg I o







20

CO CD

T J

cdT

cT

- mdash mdash 1 1 1 1 1 I - 1

Pitch -

A Roll -

A -

1 bull bull

-

1 Time (Orbits)


1 2 Time (Orbits)


gt to N

to


145h

(a) Time (Orbits)


98

1





99
























reduced


100



c n

T J

cdT




c n CD

cdT a

o T J

CQ

- - I 1 1 1 I 1 1 1 1

P i t c h

R o l l

bull

1 -

0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


E gt

10 N

CO


145h

(a) Time (Orbits)


101






102






CD

eg CQ

0 1 2 Time (Orbits)


1 2 Time (Orbits)


gt CO


145h

(a) Time (Orbits)


103





0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


-07 o 1


1E-5

0E0F

Time (Orbits)


104





o 1 2 Time (Orbits)


0 1 2 Time (Orbits)


CD CD

CM CO


to

150

100

1 2 3 0 1 2


to

(a)

50

2 3 Time (Orbits)


105





2 0 h

E 2L


0 5

1 2 3



1 2


1 2 3 Time (Orbits)


2 E - 5

1 2 3


1 2


1 E - 5

E

O E O

Time (Orbits)


106

























107





0 1 2 Time (Orbits)


CO CD

cdT

CO CD

T J CM

CO

1 2 Time (Orbits)



150

100



CO

(a)

x 50

2 3 Time (Orbits)


108




E

1 2 Time (Orbits)


2E-5

1 2 Time (Orbits)


OEOft

1 2 3 Time (Orbits)


1 2 3 Time (Orbits)


1 2 Time (Orbits)


1E-5 E z

2

0E0



109















are carried out











110












efficient












111






91 remains constant








by Ms

1 gives

qr = -Ms 1Csqr - Ms

lKsqr



Pu t t ing

v =

+

-M~lCs -M~Cd -1

0

-Mg~lMd

Id

0 v + -M~lKs -M~lKd

0 0

-M-lfs

0



(518)

(519)

x =Ax + Bud + Fd (520)

112

where

A = MC MK Id12 o

24x24

and

Let

B = ^ U 5 R 2 4 X 3 -

Fd = I FJ ) e f t 2 4 1

mdash _ i _ -5

(521)

(522)

(523)

(524)







S=Ax + Bud (526)






K(o)sx

V

dx ~ dXQ dy ~ dyQ

d z - d z0 J

Jy

dx da XQ vo

d z - dZQ ) dy dyQ

(527)

113


























114

u

MODEL UNCERTAINTY

SYSTEM PLANT

y

u LQR CONTROLLER

X KBF ESTIMATOR

u

y

LQG COMPENSATOR

PLANT

X = f v ( X t )

COMPENSATOR

x = A k x - B k y

ud

= C k f

y


115














summarized below



(528)


0 0 $u(lt) 0 0 0 0 1

0 0 1 0 0 0 0 1

(529)


components of xu


Qu =

bull1 0 0 o -0 1 0 0 0 0 1 0

0 0 0 10

(530)

116


selected as 1 0 0 0 0 1 0 0 0 0 4 0

LO 0 0 1


(531)

i o 0 15

(532)





where r is a scaler













117

100

co

gt CO

-50

-100 10


Target r = 5x10 5

r = 0 (LQG)


(a)


101 10


100

50

X3

gt 0

CO

-50

bull100 10




(b)


101 10


118


parts[63]



Dv = Cvxv


(534)

Cy

0 0 0 0 (0) 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0

h 0 0 0

0 0 0 lt ( 0 ) o 0

0 1 0 0 0 1

0 0 0

2 r

h 0 0

0 0 1 0 0 1

(535)


O f Xy


Qv

10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

(536)

mdash


1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 x 10 4 0

0 0 0 0 0 0 0 9 x 10 4

(537)

119


1 0 0 0 0 1 0 0 0 0 5 x 10 4 0

LO 0 0 5 x 10 4

(538)





(539)







these conditions



xf =

$d =

Aku

0

0 Akv

Cku o

Cu 0

Bku

0 B Xf

Xf = Ckxf

0

kv

Hf = Akxf - Bkyf

(540)

Vu Vv 0 Cv

Xf = CfXf



120


X

(541)

Ckxf















vibration case




tems

121






122





t o

1 2 Time (Orbits)



(b)

Figure 5-8



123

























124


























125











angles

















126

flexible tether






















based experiments


dynamics

127

BIBLIOGRAPHY
















128















129














130

6 1995 pp 373-384




















Letter V o l 2 No 1 1982 pp 48-52


131








132























133





w = Av (12)


4-1 ltL3gt j







equation Similarly

E = aAB + bCD (15)








notation

Wi = AjiVj (17)

134




tensor notation as

di-dx-j-1 ( L 8 )


f - a )


is given by

C = Ax)Bx) + A(x)Bx) (110)






13 Forcing Function








135


Mij = qsRv

ni^MtnrnRv

mj + Rv

niMtnTn^Rv

mj + Rv

niMtnmRv

mjs) (113)


-JPdMu 1 bdquo T x






dq dq dqt = dq^QP^




Fk(q^q^)=(RP

sk + R P n J q n ) ^ - - Q d k





(118)

136

















attitude angles

ci = on + 6

Pi = Pi (HI)

7t = 0



137



-S 0 c Pi

(II2)



(215)



(II3) Vi = I Pi i


where

and



(II4)

(II5)

(II6)

D1 = D D l D S v

52 = 31 + P(d2)m +T[d2

D3 = D2 + l2 + dX2Pii)m + T2i6x






138


KH = miDD + 2DiP






TV



d t dqred d(ired

with





139

ABSTRACT























ii















21 Kinematics 14







i i i
























iv























v





A P P E N D I C E S








v i

LIST OF SYMBOLS





respectively








true anomaly




tion








m

Pi

w 0 i

v i i






modulus E






K A Z $i(k + dXi+1)











0N) order-N




matrix

v i i i








th



respectively





in the ith link






F-Q mdash








ix




- $ T



qv---QtNT


adjacent links)



i + $DJi




tions respectively





LIST OF FIGURES


satellite system 2

























xi













without offset










without offset






x i i






















the local vertical








x i i i








xiv



















xv

1 INTRODUCTION























1


2

Satellite N


2


3



























4





tether





















5


lt mdash

lt Satellite


6


























7















ML+F = Q





be considered






8




























9



























10



























11

two methods[39]
























12





motion



modes










( L Q G L T - R )








13


21 Kinematics















frame



Di + Ti(i + f(i)) (21)

14

15


Ui nvi

wi I I i = 1

nzi bull

(22)



taken as 2 j - l

Hi(xiii)=[j) (23)




k (24)





ff(i) = $i(xili)5i(t) (25)


0 0

^i(xiJi) =


0

0 0

0

G 5R 3 x n f

(26)



16






( 3 d

dx~ + dT)^Xili^ ( 2 8 )

one arrives at











1 P i tch F[ = Cf ( a j )F 0



17

where

c f (A) =

-Sai Cai

0 0

and

C(7i ) =

0 -s0i] = 0 1 0

0 cpi

i 0 0 = 0

0

(211)

(212)

(213)








(214) SliCPi CliCPi









18



















(219)

that

(220)

19


20


cos(0i) 0 0 s i n ( 0 i ) 0 0

0 0 1

Dri

h (221)

= DTlDSv




1 1 1 1 J (222)

= D D l D 3 v


(223)



0 0 1

DTl = D D i = I d 3 i = 2N (224)


2N (225)








21





(226)





and

Ai = i(ti) - ^(t 0 -) = - ^ ( t 2 i ) (227)


V = lUli l^lL (228)

1 tf-tQ v y

A t = (229)


221 Kinetic energy




4 ^ ^ PM) T i $ 4 TiSi]i[t (231)

22

where

m

k J

x l (232)



rewritten as

1 T

(233)


Mt =



pound Sfttiqxnq

(234)




i=l (235)



(236) Mt =

0 0

0 Mt2

0

0 0

MH

0 0 0

0 0 0 bull MtN



23


Mti

m D D l


sym 0

0


0 Id nfj

0 m~

i = 1 3 5 N

(237)


gets

D D l

1 D D l = 1 0 0 0 D n

2 0 (238) 0 0 1










sym sym



tr(ffTai

TTaifi) tr(ffTai





(241)



24

and

j tr(A) =tr(^J A^j (243)



P(fi)Pii)dmi =

tr(Tai

TTaiImi) tTTai


TT7Jmi) sym sym sym

(244)


expanded is


sym sym


sym J z^dmi J


(245)






can be written as

V9i f dm f


dm

inn Lgti + T j ^ | (246)





(2-lt 7)

25






readily that



224 Strain energy












(249)




26




by





number of modes







EX^Ei+jEi (252)





experimentally [49]

27


jet = (254) wo-





(ii) Jo




dWd

^ = - i p = 2 4 ( 2 5 7 )

= 0 i = 13 N









28

system

Mq t)t+ F(q q t) = Q(q q t) (259)


















Let

e Unqxl (261) m


29











3 = 1



k=l



Qt Jfe=l

where

and


T fc-1 0 TkKAk

0 0 0 0 0 0 0 0 0 0 0 0 -

r T i - i 0 0 0 0 0 0 0 0 0 G

0 0 0 1

Qt = RpQ

G R n lt x l

(265)

(266)

G Unq x l (267)

(268)

30

where

RP

R 1 0 0 R R 2 0 R RP

2 R 3

0 0 0

R RP R PN _ 1 RN J

(269)




transformations



t h l ink as

D = A - l + Ti_iK + + K A i _ i ^ _ i


(271) (











31




t = Rvil (2-74)


Here

Rv


0 0 0

T3V nN-l

R bullN

e Mnqqxl

Ri =


e K n lt x l

RS

T i-1 0 0 o-0 0 0 0 0 0

0 0 0 1

G Mnq x l

(275)

(276)

(277)




(278)


0 0

0 0

0 0

G $lnq x l


(279)

pound = Rnjt + Rdq


32

where

and

- 0 0 0 RI 0 0

Rn = 0 0

0 0

Rf 0 0 0 rgtd K2 0

Rd = 0 0 Rl

0 0 0

RN-1 0

G 5R n w x l

0 0 0 e 5 R N ^ X L

(281)

(282)






where

D D T i


DTL 0 0 0 d 3 0 0 0 ID

0 0 nfj 0

RP =


2 R3

iJfDTTi i^

x l

0 0 0

R PN-I RN

(284)

(285)



33



0 0 0 0 0 0 0 0

(286)









M = RV MtRv


(289)

(290)

Inverting Eq(290)



34


Mr1

h 0 0

0 0

0 0

0 0 0

0

0

0

tN

(292)



operation








Mq + f = P C A (293)




pcTM-pc

= ( dc + PdegTM-^f) (294)




35


















nu ^5 Qk=E Pern ^ (2-97)

m=l deg Q k






36






Defining


= d + dfY + T 3 f c m 3



(298)

(299)



(2100)

as i - 2

k=l i1 cm (2101)



WjRi A R i (2102)




1 Q

Q3

3 -1

T 3 T t 2 (2103)

37

Satellite 3

mdash gt -raquo cm

D1=d1

mdash gt

a mdash gt

o


38




Q

~Qgt1

0

0

( T ^ 2 ) (2104)







(2105)

where





3

Qk = ^ F e i

i=l

d

dqk (2106)




39


40

be written as


QiMmi



(2107)

(2108)



Q

Q T 3 0 Q T 5 0

0 0 0

0 lt33T3 Qyen Q5

3T5 0

0 0 0 Q4T5 0

0 0 0 Q5

5T5

M M I

Ti 2 m 3

Quu (2109)


41










numerically














42















by letting

thus

X - ^ ^ f ^ ) =ltbullbull) (3 -2 )







43








plott ing package



















44

INIT

DGEAR PARAM

IC amp SYS PARAM

MAIN

lt

DGEAR

lt

FCN

OUTPUT



VIBRATION

r OFFS DYNA

ET MICS


45









longer stored




the system











46

h =

1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760



0 400 0 0 0 400












fcm3 = 10 0T










47




OOEOh

-50E-12

-10E-11 I i i



48



(a)

20E6

10E6r-

00E0

-10E6

-20E6


-

1 I I I 1 I I

I I

AP e ~_

i i 1 i i i i

00E0

UJ -50E-12 U J

lt


-10E-11h

(b) Time (Orbits)


49


























50

a p(0) = 2deg 0(0) = 2C

6(0) = 08 m

0(0) = 001 m


D p y = 0 D p 2 = 0

a oo

c -ooo -0 01


51






^MAAAAAAAWVWWWWVVVW

V bull i bull i I i J 00 05 10 15 20


Time (Orbits)


52
























53


1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760

h =

h = 200 0 0

0 400 0 0 0 400















mdash







54



55




























56







57







(b) Time (Orbits)


58










by the coupling

















59








60



Time (Orbits)



Time (Orbits)



0 1 2 3 4 5 (b) Time (Orbits)


61





Tether Pitch Angle


Tether Roll Angle

CD

edT

1 2 3 Time (Orbits)




CD

C O

(a)

Figure 4-




62



0 1 2 3 4 5







0 1 2 3 4 5 (b) Time (Orbits)


63



2 3 4 Time (Orbits)

Tether Pitch Angle


Tether Roll Angle



2 3 4 Time (Orbits)


(a)

Figure 4-


2 3 4 Time (Orbits)


64




(b) Time (Orbits)


65










V0 = 146 ms
















66



i i i i i i i i i i i 0 1 2 3 4 5 0 1 2 3 4 5






67



0 1 2 3 4 5


(b) Time (Orbits)


68




E





Tether Pitch Angle


Tether Roll Angle

CD

oo ax


_ 1 _ L





-50

(a) 1 2 3 4


Time (Orbits)


69




to

-02

(b)

Figure 4-7

2 3 Time (Orbits)


0 1 1 bull 1 1



5

1 1 1 1 1 i i | i i | ^

_ i i i i L -I



70












roll

45 Tether Retrieval










and subsatellite



71




E


1 2 3 Time (Orbits)




Tether Pitch Angle

1 2 3 Time (Orbits)

Tether Roll Angle

i i i i J I I i _





D )

T J ^ 0 E 0 eg

CQ

-3E-3

(a) 1 2 3 4


Time (Orbits)


7 2





n 1 1 1 f

I I I _j I I I I I 1 1 1 1 1 2 3 4 5



Time (Orbits)


(b)

Figure 4-8

73




(a) Time (Orbits)



i i i i i i i i i i I 0 1 2 3






04- -



74




E 10 to


1 2 Time (Orbits)


(b) Time (Orbits)


75

























76


77



o 1 2 Time (Orbits)


0 1 2 Time (Orbits)






78


p1(0)=p2(0)=(33(0)=pt1(0)=pt2(0)=10





79



























80






i i i i i 0 1 2 3 4 5










81






82



























83



I- I i t i i i i I H 0 1 2 0 1 2





84



laquo = 5 E - 3 |

000 002



Time (Orbits)


85




E

2000

cu T3

2000

co CD

2000

C D CD

T J


1 2 Time (Orbits)


cn CD

33 cdl




1 2 Time (Orbits)




86



5E-3h

co

0E0

025

000 E to

-025

002

-002

(b)

Figure 4-14




Time (Orbits)


87






















88





89


1E-2

to

OEO

E 00

to


005



Time (Orbits)


90





91




92

























93


present case













mance











94





by Eq(2109)

Lett ing

~fr = fr~ QuU (52)


ir = s(frMrdc) (53)



mdash = Fr - Qru





Ire = Frc ~~ Qrcu

(56)

mdash vrci






an(^ qrcd

95









-In ) 0 5 V T ^ r (59)


V (510)








qrcd 37Vxl (511)

96


e U 3 N x l (512)

E K 3 7 V x l (513)

















97

and

o 0

Qrcj mdash

o o

0 0

Oiit) 0 0

1

N

N

1deg I o







20

CO CD

T J

cdT

cT

- mdash mdash 1 1 1 1 1 I - 1

Pitch -

A Roll -

A -

1 bull bull

-

1 Time (Orbits)


1 2 Time (Orbits)


gt to N

to


145h

(a) Time (Orbits)


98

1





99
























reduced


100



c n

T J

cdT




c n CD

cdT a

o T J

CQ

- - I 1 1 1 I 1 1 1 1

P i t c h

R o l l

bull

1 -

0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


E gt

10 N

CO


145h

(a) Time (Orbits)


101






102






CD

eg CQ

0 1 2 Time (Orbits)


1 2 Time (Orbits)


gt CO


145h

(a) Time (Orbits)


103





0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


-07 o 1


1E-5

0E0F

Time (Orbits)


104





o 1 2 Time (Orbits)


0 1 2 Time (Orbits)


CD CD

CM CO


to

150

100

1 2 3 0 1 2


to

(a)

50

2 3 Time (Orbits)


105





2 0 h

E 2L


0 5

1 2 3



1 2


1 2 3 Time (Orbits)


2 E - 5

1 2 3


1 2


1 E - 5

E

O E O

Time (Orbits)


106

























107





0 1 2 Time (Orbits)


CO CD

cdT

CO CD

T J CM

CO

1 2 Time (Orbits)



150

100



CO

(a)

x 50

2 3 Time (Orbits)


108




E

1 2 Time (Orbits)


2E-5

1 2 Time (Orbits)


OEOft

1 2 3 Time (Orbits)


1 2 3 Time (Orbits)


1 2 Time (Orbits)


1E-5 E z

2

0E0



109















are carried out











110












efficient












111






91 remains constant








by Ms

1 gives

qr = -Ms 1Csqr - Ms

lKsqr



Pu t t ing

v =

+

-M~lCs -M~Cd -1

0

-Mg~lMd

Id

0 v + -M~lKs -M~lKd

0 0

-M-lfs

0



(518)

(519)

x =Ax + Bud + Fd (520)

112

where

A = MC MK Id12 o

24x24

and

Let

B = ^ U 5 R 2 4 X 3 -

Fd = I FJ ) e f t 2 4 1

mdash _ i _ -5

(521)

(522)

(523)

(524)







S=Ax + Bud (526)






K(o)sx

V

dx ~ dXQ dy ~ dyQ

d z - d z0 J

Jy

dx da XQ vo

d z - dZQ ) dy dyQ

(527)

113


























114

u

MODEL UNCERTAINTY

SYSTEM PLANT

y

u LQR CONTROLLER

X KBF ESTIMATOR

u

y

LQG COMPENSATOR

PLANT

X = f v ( X t )

COMPENSATOR

x = A k x - B k y

ud

= C k f

y


115














summarized below



(528)


0 0 $u(lt) 0 0 0 0 1

0 0 1 0 0 0 0 1

(529)


components of xu


Qu =

bull1 0 0 o -0 1 0 0 0 0 1 0

0 0 0 10

(530)

116


selected as 1 0 0 0 0 1 0 0 0 0 4 0

LO 0 0 1


(531)

i o 0 15

(532)





where r is a scaler













117

100

co

gt CO

-50

-100 10


Target r = 5x10 5

r = 0 (LQG)


(a)


101 10


100

50

X3

gt 0

CO

-50

bull100 10




(b)


101 10


118


parts[63]



Dv = Cvxv


(534)

Cy

0 0 0 0 (0) 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0

h 0 0 0

0 0 0 lt ( 0 ) o 0

0 1 0 0 0 1

0 0 0

2 r

h 0 0

0 0 1 0 0 1

(535)


O f Xy


Qv

10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

(536)

mdash


1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 x 10 4 0

0 0 0 0 0 0 0 9 x 10 4

(537)

119


1 0 0 0 0 1 0 0 0 0 5 x 10 4 0

LO 0 0 5 x 10 4

(538)





(539)







these conditions



xf =

$d =

Aku

0

0 Akv

Cku o

Cu 0

Bku

0 B Xf

Xf = Ckxf

0

kv

Hf = Akxf - Bkyf

(540)

Vu Vv 0 Cv

Xf = CfXf



120


X

(541)

Ckxf















vibration case




tems

121






122





t o

1 2 Time (Orbits)



(b)

Figure 5-8



123

























124


























125











angles

















126

flexible tether






















based experiments


dynamics

127

BIBLIOGRAPHY
















128















129














130

6 1995 pp 373-384




















Letter V o l 2 No 1 1982 pp 48-52


131








132























133





w = Av (12)


4-1 ltL3gt j







equation Similarly

E = aAB + bCD (15)








notation

Wi = AjiVj (17)

134




tensor notation as

di-dx-j-1 ( L 8 )


f - a )


is given by

C = Ax)Bx) + A(x)Bx) (110)






13 Forcing Function








135


Mij = qsRv

ni^MtnrnRv

mj + Rv

niMtnTn^Rv

mj + Rv

niMtnmRv

mjs) (113)


-JPdMu 1 bdquo T x






dq dq dqt = dq^QP^




Fk(q^q^)=(RP

sk + R P n J q n ) ^ - - Q d k





(118)

136

















attitude angles

ci = on + 6

Pi = Pi (HI)

7t = 0



137



-S 0 c Pi

(II2)



(215)



(II3) Vi = I Pi i


where

and



(II4)

(II5)

(II6)

D1 = D D l D S v

52 = 31 + P(d2)m +T[d2

D3 = D2 + l2 + dX2Pii)m + T2i6x






138


KH = miDD + 2DiP






TV



d t dqred d(ired

with





139















21 Kinematics 14







i i i
























iv























v





A P P E N D I C E S








v i

LIST OF SYMBOLS





respectively








true anomaly




tion








m

Pi

w 0 i

v i i






modulus E






K A Z $i(k + dXi+1)











0N) order-N




matrix

v i i i








th



respectively





in the ith link






F-Q mdash








ix




- $ T



qv---QtNT


adjacent links)



i + $DJi




tions respectively





LIST OF FIGURES


satellite system 2

























xi













without offset










without offset






x i i






















the local vertical








x i i i








xiv



















xv

1 INTRODUCTION























1


2

Satellite N


2


3



























4





tether





















5


lt mdash

lt Satellite


6


























7















ML+F = Q





be considered






8




























9



























10



























11

two methods[39]
























12





motion



modes










( L Q G L T - R )








13


21 Kinematics















frame



Di + Ti(i + f(i)) (21)

14

15


Ui nvi

wi I I i = 1

nzi bull

(22)



taken as 2 j - l

Hi(xiii)=[j) (23)




k (24)





ff(i) = $i(xili)5i(t) (25)


0 0

^i(xiJi) =


0

0 0

0

G 5R 3 x n f

(26)



16






( 3 d

dx~ + dT)^Xili^ ( 2 8 )

one arrives at











1 P i tch F[ = Cf ( a j )F 0



17

where

c f (A) =

-Sai Cai

0 0

and

C(7i ) =

0 -s0i] = 0 1 0

0 cpi

i 0 0 = 0

0

(211)

(212)

(213)








(214) SliCPi CliCPi









18



















(219)

that

(220)

19


20


cos(0i) 0 0 s i n ( 0 i ) 0 0

0 0 1

Dri

h (221)

= DTlDSv




1 1 1 1 J (222)

= D D l D 3 v


(223)



0 0 1

DTl = D D i = I d 3 i = 2N (224)


2N (225)








21





(226)





and

Ai = i(ti) - ^(t 0 -) = - ^ ( t 2 i ) (227)


V = lUli l^lL (228)

1 tf-tQ v y

A t = (229)


221 Kinetic energy




4 ^ ^ PM) T i $ 4 TiSi]i[t (231)

22

where

m

k J

x l (232)



rewritten as

1 T

(233)


Mt =



pound Sfttiqxnq

(234)




i=l (235)



(236) Mt =

0 0

0 Mt2

0

0 0

MH

0 0 0

0 0 0 bull MtN



23


Mti

m D D l


sym 0

0


0 Id nfj

0 m~

i = 1 3 5 N

(237)


gets

D D l

1 D D l = 1 0 0 0 D n

2 0 (238) 0 0 1










sym sym



tr(ffTai

TTaifi) tr(ffTai





(241)



24

and

j tr(A) =tr(^J A^j (243)



P(fi)Pii)dmi =

tr(Tai

TTaiImi) tTTai


TT7Jmi) sym sym sym

(244)


expanded is


sym sym


sym J z^dmi J


(245)






can be written as

V9i f dm f


dm

inn Lgti + T j ^ | (246)





(2-lt 7)

25






readily that



224 Strain energy












(249)




26




by





number of modes







EX^Ei+jEi (252)





experimentally [49]

27


jet = (254) wo-





(ii) Jo




dWd

^ = - i p = 2 4 ( 2 5 7 )

= 0 i = 13 N









28

system

Mq t)t+ F(q q t) = Q(q q t) (259)


















Let

e Unqxl (261) m


29











3 = 1



k=l



Qt Jfe=l

where

and


T fc-1 0 TkKAk

0 0 0 0 0 0 0 0 0 0 0 0 -

r T i - i 0 0 0 0 0 0 0 0 0 G

0 0 0 1

Qt = RpQ

G R n lt x l

(265)

(266)

G Unq x l (267)

(268)

30

where

RP

R 1 0 0 R R 2 0 R RP

2 R 3

0 0 0

R RP R PN _ 1 RN J

(269)




transformations



t h l ink as

D = A - l + Ti_iK + + K A i _ i ^ _ i


(271) (











31




t = Rvil (2-74)


Here

Rv


0 0 0

T3V nN-l

R bullN

e Mnqqxl

Ri =


e K n lt x l

RS

T i-1 0 0 o-0 0 0 0 0 0

0 0 0 1

G Mnq x l

(275)

(276)

(277)




(278)


0 0

0 0

0 0

G $lnq x l


(279)

pound = Rnjt + Rdq


32

where

and

- 0 0 0 RI 0 0

Rn = 0 0

0 0

Rf 0 0 0 rgtd K2 0

Rd = 0 0 Rl

0 0 0

RN-1 0

G 5R n w x l

0 0 0 e 5 R N ^ X L

(281)

(282)






where

D D T i


DTL 0 0 0 d 3 0 0 0 ID

0 0 nfj 0

RP =


2 R3

iJfDTTi i^

x l

0 0 0

R PN-I RN

(284)

(285)



33



0 0 0 0 0 0 0 0

(286)









M = RV MtRv


(289)

(290)

Inverting Eq(290)



34


Mr1

h 0 0

0 0

0 0

0 0 0

0

0

0

tN

(292)



operation








Mq + f = P C A (293)




pcTM-pc

= ( dc + PdegTM-^f) (294)




35


















nu ^5 Qk=E Pern ^ (2-97)

m=l deg Q k






36






Defining


= d + dfY + T 3 f c m 3



(298)

(299)



(2100)

as i - 2

k=l i1 cm (2101)



WjRi A R i (2102)




1 Q

Q3

3 -1

T 3 T t 2 (2103)

37

Satellite 3

mdash gt -raquo cm

D1=d1

mdash gt

a mdash gt

o


38




Q

~Qgt1

0

0

( T ^ 2 ) (2104)







(2105)

where





3

Qk = ^ F e i

i=l

d

dqk (2106)




39


40

be written as


QiMmi



(2107)

(2108)



Q

Q T 3 0 Q T 5 0

0 0 0

0 lt33T3 Qyen Q5

3T5 0

0 0 0 Q4T5 0

0 0 0 Q5

5T5

M M I

Ti 2 m 3

Quu (2109)


41










numerically














42















by letting

thus

X - ^ ^ f ^ ) =ltbullbull) (3 -2 )







43








plott ing package



















44

INIT

DGEAR PARAM

IC amp SYS PARAM

MAIN

lt

DGEAR

lt

FCN

OUTPUT



VIBRATION

r OFFS DYNA

ET MICS


45









longer stored




the system











46

h =

1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760



0 400 0 0 0 400












fcm3 = 10 0T










47




OOEOh

-50E-12

-10E-11 I i i



48



(a)

20E6

10E6r-

00E0

-10E6

-20E6


-

1 I I I 1 I I

I I

AP e ~_

i i 1 i i i i

00E0

UJ -50E-12 U J

lt


-10E-11h

(b) Time (Orbits)


49


























50

a p(0) = 2deg 0(0) = 2C

6(0) = 08 m

0(0) = 001 m


D p y = 0 D p 2 = 0

a oo

c -ooo -0 01


51






^MAAAAAAAWVWWWWVVVW

V bull i bull i I i J 00 05 10 15 20


Time (Orbits)


52
























53


1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760

h =

h = 200 0 0

0 400 0 0 0 400















mdash







54



55




























56







57







(b) Time (Orbits)


58










by the coupling

















59








60



Time (Orbits)



Time (Orbits)



0 1 2 3 4 5 (b) Time (Orbits)


61





Tether Pitch Angle


Tether Roll Angle

CD

edT

1 2 3 Time (Orbits)




CD

C O

(a)

Figure 4-




62



0 1 2 3 4 5







0 1 2 3 4 5 (b) Time (Orbits)


63



2 3 4 Time (Orbits)

Tether Pitch Angle


Tether Roll Angle



2 3 4 Time (Orbits)


(a)

Figure 4-


2 3 4 Time (Orbits)


64




(b) Time (Orbits)


65










V0 = 146 ms
















66



i i i i i i i i i i i 0 1 2 3 4 5 0 1 2 3 4 5






67



0 1 2 3 4 5


(b) Time (Orbits)


68




E





Tether Pitch Angle


Tether Roll Angle

CD

oo ax


_ 1 _ L





-50

(a) 1 2 3 4


Time (Orbits)


69




to

-02

(b)

Figure 4-7

2 3 Time (Orbits)


0 1 1 bull 1 1



5

1 1 1 1 1 i i | i i | ^

_ i i i i L -I



70












roll

45 Tether Retrieval










and subsatellite



71




E


1 2 3 Time (Orbits)




Tether Pitch Angle

1 2 3 Time (Orbits)

Tether Roll Angle

i i i i J I I i _





D )

T J ^ 0 E 0 eg

CQ

-3E-3

(a) 1 2 3 4


Time (Orbits)


7 2





n 1 1 1 f

I I I _j I I I I I 1 1 1 1 1 2 3 4 5



Time (Orbits)


(b)

Figure 4-8

73




(a) Time (Orbits)



i i i i i i i i i i I 0 1 2 3






04- -



74




E 10 to


1 2 Time (Orbits)


(b) Time (Orbits)


75

























76


77



o 1 2 Time (Orbits)


0 1 2 Time (Orbits)






78


p1(0)=p2(0)=(33(0)=pt1(0)=pt2(0)=10





79



























80






i i i i i 0 1 2 3 4 5










81






82



























83



I- I i t i i i i I H 0 1 2 0 1 2





84



laquo = 5 E - 3 |

000 002



Time (Orbits)


85




E

2000

cu T3

2000

co CD

2000

C D CD

T J


1 2 Time (Orbits)


cn CD

33 cdl




1 2 Time (Orbits)




86



5E-3h

co

0E0

025

000 E to

-025

002

-002

(b)

Figure 4-14




Time (Orbits)


87






















88





89


1E-2

to

OEO

E 00

to


005



Time (Orbits)


90





91




92

























93


present case













mance











94





by Eq(2109)

Lett ing

~fr = fr~ QuU (52)


ir = s(frMrdc) (53)



mdash = Fr - Qru





Ire = Frc ~~ Qrcu

(56)

mdash vrci






an(^ qrcd

95









-In ) 0 5 V T ^ r (59)


V (510)








qrcd 37Vxl (511)

96


e U 3 N x l (512)

E K 3 7 V x l (513)

















97

and

o 0

Qrcj mdash

o o

0 0

Oiit) 0 0

1

N

N

1deg I o







20

CO CD

T J

cdT

cT

- mdash mdash 1 1 1 1 1 I - 1

Pitch -

A Roll -

A -

1 bull bull

-

1 Time (Orbits)


1 2 Time (Orbits)


gt to N

to


145h

(a) Time (Orbits)


98

1





99
























reduced


100



c n

T J

cdT




c n CD

cdT a

o T J

CQ

- - I 1 1 1 I 1 1 1 1

P i t c h

R o l l

bull

1 -

0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


E gt

10 N

CO


145h

(a) Time (Orbits)


101






102






CD

eg CQ

0 1 2 Time (Orbits)


1 2 Time (Orbits)


gt CO


145h

(a) Time (Orbits)


103





0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


-07 o 1


1E-5

0E0F

Time (Orbits)


104





o 1 2 Time (Orbits)


0 1 2 Time (Orbits)


CD CD

CM CO


to

150

100

1 2 3 0 1 2


to

(a)

50

2 3 Time (Orbits)


105





2 0 h

E 2L


0 5

1 2 3



1 2


1 2 3 Time (Orbits)


2 E - 5

1 2 3


1 2


1 E - 5

E

O E O

Time (Orbits)


106

























107





0 1 2 Time (Orbits)


CO CD

cdT

CO CD

T J CM

CO

1 2 Time (Orbits)



150

100



CO

(a)

x 50

2 3 Time (Orbits)


108




E

1 2 Time (Orbits)


2E-5

1 2 Time (Orbits)


OEOft

1 2 3 Time (Orbits)


1 2 3 Time (Orbits)


1 2 Time (Orbits)


1E-5 E z

2

0E0



109















are carried out











110












efficient












111






91 remains constant








by Ms

1 gives

qr = -Ms 1Csqr - Ms

lKsqr



Pu t t ing

v =

+

-M~lCs -M~Cd -1

0

-Mg~lMd

Id

0 v + -M~lKs -M~lKd

0 0

-M-lfs

0



(518)

(519)

x =Ax + Bud + Fd (520)

112

where

A = MC MK Id12 o

24x24

and

Let

B = ^ U 5 R 2 4 X 3 -

Fd = I FJ ) e f t 2 4 1

mdash _ i _ -5

(521)

(522)

(523)

(524)







S=Ax + Bud (526)






K(o)sx

V

dx ~ dXQ dy ~ dyQ

d z - d z0 J

Jy

dx da XQ vo

d z - dZQ ) dy dyQ

(527)

113


























114

u

MODEL UNCERTAINTY

SYSTEM PLANT

y

u LQR CONTROLLER

X KBF ESTIMATOR

u

y

LQG COMPENSATOR

PLANT

X = f v ( X t )

COMPENSATOR

x = A k x - B k y

ud

= C k f

y


115














summarized below



(528)


0 0 $u(lt) 0 0 0 0 1

0 0 1 0 0 0 0 1

(529)


components of xu


Qu =

bull1 0 0 o -0 1 0 0 0 0 1 0

0 0 0 10

(530)

116


selected as 1 0 0 0 0 1 0 0 0 0 4 0

LO 0 0 1


(531)

i o 0 15

(532)





where r is a scaler













117

100

co

gt CO

-50

-100 10


Target r = 5x10 5

r = 0 (LQG)


(a)


101 10


100

50

X3

gt 0

CO

-50

bull100 10




(b)


101 10


118


parts[63]



Dv = Cvxv


(534)

Cy

0 0 0 0 (0) 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0

h 0 0 0

0 0 0 lt ( 0 ) o 0

0 1 0 0 0 1

0 0 0

2 r

h 0 0

0 0 1 0 0 1

(535)


O f Xy


Qv

10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

(536)

mdash


1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 x 10 4 0

0 0 0 0 0 0 0 9 x 10 4

(537)

119


1 0 0 0 0 1 0 0 0 0 5 x 10 4 0

LO 0 0 5 x 10 4

(538)





(539)







these conditions



xf =

$d =

Aku

0

0 Akv

Cku o

Cu 0

Bku

0 B Xf

Xf = Ckxf

0

kv

Hf = Akxf - Bkyf

(540)

Vu Vv 0 Cv

Xf = CfXf



120


X

(541)

Ckxf















vibration case




tems

121






122





t o

1 2 Time (Orbits)



(b)

Figure 5-8



123

























124


























125











angles

















126

flexible tether






















based experiments


dynamics

127

BIBLIOGRAPHY
















128















129














130

6 1995 pp 373-384




















Letter V o l 2 No 1 1982 pp 48-52


131








132























133





w = Av (12)


4-1 ltL3gt j







equation Similarly

E = aAB + bCD (15)








notation

Wi = AjiVj (17)

134




tensor notation as

di-dx-j-1 ( L 8 )


f - a )


is given by

C = Ax)Bx) + A(x)Bx) (110)






13 Forcing Function








135


Mij = qsRv

ni^MtnrnRv

mj + Rv

niMtnTn^Rv

mj + Rv

niMtnmRv

mjs) (113)


-JPdMu 1 bdquo T x






dq dq dqt = dq^QP^




Fk(q^q^)=(RP

sk + R P n J q n ) ^ - - Q d k





(118)

136

















attitude angles

ci = on + 6

Pi = Pi (HI)

7t = 0



137



-S 0 c Pi

(II2)



(215)



(II3) Vi = I Pi i


where

and



(II4)

(II5)

(II6)

D1 = D D l D S v

52 = 31 + P(d2)m +T[d2

D3 = D2 + l2 + dX2Pii)m + T2i6x






138


KH = miDD + 2DiP






TV



d t dqred d(ired

with





139
























iv























v





A P P E N D I C E S








v i

LIST OF SYMBOLS





respectively








true anomaly




tion








m

Pi

w 0 i

v i i






modulus E






K A Z $i(k + dXi+1)











0N) order-N




matrix

v i i i








th



respectively





in the ith link






F-Q mdash








ix




- $ T



qv---QtNT


adjacent links)



i + $DJi




tions respectively





LIST OF FIGURES


satellite system 2

























xi













without offset










without offset






x i i






















the local vertical








x i i i








xiv



















xv

1 INTRODUCTION























1


2

Satellite N


2


3



























4





tether





















5


lt mdash

lt Satellite


6


























7















ML+F = Q





be considered






8




























9



























10



























11

two methods[39]
























12





motion



modes










( L Q G L T - R )








13


21 Kinematics















frame



Di + Ti(i + f(i)) (21)

14

15


Ui nvi

wi I I i = 1

nzi bull

(22)



taken as 2 j - l

Hi(xiii)=[j) (23)




k (24)





ff(i) = $i(xili)5i(t) (25)


0 0

^i(xiJi) =


0

0 0

0

G 5R 3 x n f

(26)



16






( 3 d

dx~ + dT)^Xili^ ( 2 8 )

one arrives at











1 P i tch F[ = Cf ( a j )F 0



17

where

c f (A) =

-Sai Cai

0 0

and

C(7i ) =

0 -s0i] = 0 1 0

0 cpi

i 0 0 = 0

0

(211)

(212)

(213)








(214) SliCPi CliCPi









18



















(219)

that

(220)

19


20


cos(0i) 0 0 s i n ( 0 i ) 0 0

0 0 1

Dri

h (221)

= DTlDSv




1 1 1 1 J (222)

= D D l D 3 v


(223)



0 0 1

DTl = D D i = I d 3 i = 2N (224)


2N (225)








21





(226)





and

Ai = i(ti) - ^(t 0 -) = - ^ ( t 2 i ) (227)


V = lUli l^lL (228)

1 tf-tQ v y

A t = (229)


221 Kinetic energy




4 ^ ^ PM) T i $ 4 TiSi]i[t (231)

22

where

m

k J

x l (232)



rewritten as

1 T

(233)


Mt =



pound Sfttiqxnq

(234)




i=l (235)



(236) Mt =

0 0

0 Mt2

0

0 0

MH

0 0 0

0 0 0 bull MtN



23


Mti

m D D l


sym 0

0


0 Id nfj

0 m~

i = 1 3 5 N

(237)


gets

D D l

1 D D l = 1 0 0 0 D n

2 0 (238) 0 0 1










sym sym



tr(ffTai

TTaifi) tr(ffTai





(241)



24

and

j tr(A) =tr(^J A^j (243)



P(fi)Pii)dmi =

tr(Tai

TTaiImi) tTTai


TT7Jmi) sym sym sym

(244)


expanded is


sym sym


sym J z^dmi J


(245)






can be written as

V9i f dm f


dm

inn Lgti + T j ^ | (246)





(2-lt 7)

25






readily that



224 Strain energy












(249)




26




by





number of modes







EX^Ei+jEi (252)





experimentally [49]

27


jet = (254) wo-





(ii) Jo




dWd

^ = - i p = 2 4 ( 2 5 7 )

= 0 i = 13 N









28

system

Mq t)t+ F(q q t) = Q(q q t) (259)


















Let

e Unqxl (261) m


29











3 = 1



k=l



Qt Jfe=l

where

and


T fc-1 0 TkKAk

0 0 0 0 0 0 0 0 0 0 0 0 -

r T i - i 0 0 0 0 0 0 0 0 0 G

0 0 0 1

Qt = RpQ

G R n lt x l

(265)

(266)

G Unq x l (267)

(268)

30

where

RP

R 1 0 0 R R 2 0 R RP

2 R 3

0 0 0

R RP R PN _ 1 RN J

(269)




transformations



t h l ink as

D = A - l + Ti_iK + + K A i _ i ^ _ i


(271) (











31




t = Rvil (2-74)


Here

Rv


0 0 0

T3V nN-l

R bullN

e Mnqqxl

Ri =


e K n lt x l

RS

T i-1 0 0 o-0 0 0 0 0 0

0 0 0 1

G Mnq x l

(275)

(276)

(277)




(278)


0 0

0 0

0 0

G $lnq x l


(279)

pound = Rnjt + Rdq


32

where

and

- 0 0 0 RI 0 0

Rn = 0 0

0 0

Rf 0 0 0 rgtd K2 0

Rd = 0 0 Rl

0 0 0

RN-1 0

G 5R n w x l

0 0 0 e 5 R N ^ X L

(281)

(282)






where

D D T i


DTL 0 0 0 d 3 0 0 0 ID

0 0 nfj 0

RP =


2 R3

iJfDTTi i^

x l

0 0 0

R PN-I RN

(284)

(285)



33



0 0 0 0 0 0 0 0

(286)









M = RV MtRv


(289)

(290)

Inverting Eq(290)



34


Mr1

h 0 0

0 0

0 0

0 0 0

0

0

0

tN

(292)



operation








Mq + f = P C A (293)




pcTM-pc

= ( dc + PdegTM-^f) (294)




35


















nu ^5 Qk=E Pern ^ (2-97)

m=l deg Q k






36






Defining


= d + dfY + T 3 f c m 3



(298)

(299)



(2100)

as i - 2

k=l i1 cm (2101)



WjRi A R i (2102)




1 Q

Q3

3 -1

T 3 T t 2 (2103)

37

Satellite 3

mdash gt -raquo cm

D1=d1

mdash gt

a mdash gt

o


38




Q

~Qgt1

0

0

( T ^ 2 ) (2104)







(2105)

where





3

Qk = ^ F e i

i=l

d

dqk (2106)




39


40

be written as


QiMmi



(2107)

(2108)



Q

Q T 3 0 Q T 5 0

0 0 0

0 lt33T3 Qyen Q5

3T5 0

0 0 0 Q4T5 0

0 0 0 Q5

5T5

M M I

Ti 2 m 3

Quu (2109)


41










numerically














42















by letting

thus

X - ^ ^ f ^ ) =ltbullbull) (3 -2 )







43








plott ing package



















44

INIT

DGEAR PARAM

IC amp SYS PARAM

MAIN

lt

DGEAR

lt

FCN

OUTPUT



VIBRATION

r OFFS DYNA

ET MICS


45









longer stored




the system











46

h =

1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760



0 400 0 0 0 400












fcm3 = 10 0T










47




OOEOh

-50E-12

-10E-11 I i i



48



(a)

20E6

10E6r-

00E0

-10E6

-20E6


-

1 I I I 1 I I

I I

AP e ~_

i i 1 i i i i

00E0

UJ -50E-12 U J

lt


-10E-11h

(b) Time (Orbits)


49


























50

a p(0) = 2deg 0(0) = 2C

6(0) = 08 m

0(0) = 001 m


D p y = 0 D p 2 = 0

a oo

c -ooo -0 01


51






^MAAAAAAAWVWWWWVVVW

V bull i bull i I i J 00 05 10 15 20


Time (Orbits)


52
























53


1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760

h =

h = 200 0 0

0 400 0 0 0 400















mdash







54



55




























56







57







(b) Time (Orbits)


58










by the coupling

















59








60



Time (Orbits)



Time (Orbits)



0 1 2 3 4 5 (b) Time (Orbits)


61





Tether Pitch Angle


Tether Roll Angle

CD

edT

1 2 3 Time (Orbits)




CD

C O

(a)

Figure 4-




62



0 1 2 3 4 5







0 1 2 3 4 5 (b) Time (Orbits)


63



2 3 4 Time (Orbits)

Tether Pitch Angle


Tether Roll Angle



2 3 4 Time (Orbits)


(a)

Figure 4-


2 3 4 Time (Orbits)


64




(b) Time (Orbits)


65










V0 = 146 ms
















66



i i i i i i i i i i i 0 1 2 3 4 5 0 1 2 3 4 5






67



0 1 2 3 4 5


(b) Time (Orbits)


68




E





Tether Pitch Angle


Tether Roll Angle

CD

oo ax


_ 1 _ L





-50

(a) 1 2 3 4


Time (Orbits)


69




to

-02

(b)

Figure 4-7

2 3 Time (Orbits)


0 1 1 bull 1 1



5

1 1 1 1 1 i i | i i | ^

_ i i i i L -I



70












roll

45 Tether Retrieval










and subsatellite



71




E


1 2 3 Time (Orbits)




Tether Pitch Angle

1 2 3 Time (Orbits)

Tether Roll Angle

i i i i J I I i _





D )

T J ^ 0 E 0 eg

CQ

-3E-3

(a) 1 2 3 4


Time (Orbits)


7 2





n 1 1 1 f

I I I _j I I I I I 1 1 1 1 1 2 3 4 5



Time (Orbits)


(b)

Figure 4-8

73




(a) Time (Orbits)



i i i i i i i i i i I 0 1 2 3






04- -



74




E 10 to


1 2 Time (Orbits)


(b) Time (Orbits)


75

























76


77



o 1 2 Time (Orbits)


0 1 2 Time (Orbits)






78


p1(0)=p2(0)=(33(0)=pt1(0)=pt2(0)=10





79



























80






i i i i i 0 1 2 3 4 5










81






82



























83



I- I i t i i i i I H 0 1 2 0 1 2





84



laquo = 5 E - 3 |

000 002



Time (Orbits)


85




E

2000

cu T3

2000

co CD

2000

C D CD

T J


1 2 Time (Orbits)


cn CD

33 cdl




1 2 Time (Orbits)




86



5E-3h

co

0E0

025

000 E to

-025

002

-002

(b)

Figure 4-14




Time (Orbits)


87






















88





89


1E-2

to

OEO

E 00

to


005



Time (Orbits)


90





91




92

























93


present case













mance











94





by Eq(2109)

Lett ing

~fr = fr~ QuU (52)


ir = s(frMrdc) (53)



mdash = Fr - Qru





Ire = Frc ~~ Qrcu

(56)

mdash vrci






an(^ qrcd

95









-In ) 0 5 V T ^ r (59)


V (510)








qrcd 37Vxl (511)

96


e U 3 N x l (512)

E K 3 7 V x l (513)

















97

and

o 0

Qrcj mdash

o o

0 0

Oiit) 0 0

1

N

N

1deg I o







20

CO CD

T J

cdT

cT

- mdash mdash 1 1 1 1 1 I - 1

Pitch -

A Roll -

A -

1 bull bull

-

1 Time (Orbits)


1 2 Time (Orbits)


gt to N

to


145h

(a) Time (Orbits)


98

1





99
























reduced


100



c n

T J

cdT




c n CD

cdT a

o T J

CQ

- - I 1 1 1 I 1 1 1 1

P i t c h

R o l l

bull

1 -

0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


E gt

10 N

CO


145h

(a) Time (Orbits)


101






102






CD

eg CQ

0 1 2 Time (Orbits)


1 2 Time (Orbits)


gt CO


145h

(a) Time (Orbits)


103





0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


-07 o 1


1E-5

0E0F

Time (Orbits)


104





o 1 2 Time (Orbits)


0 1 2 Time (Orbits)


CD CD

CM CO


to

150

100

1 2 3 0 1 2


to

(a)

50

2 3 Time (Orbits)


105





2 0 h

E 2L


0 5

1 2 3



1 2


1 2 3 Time (Orbits)


2 E - 5

1 2 3


1 2


1 E - 5

E

O E O

Time (Orbits)


106

























107





0 1 2 Time (Orbits)


CO CD

cdT

CO CD

T J CM

CO

1 2 Time (Orbits)



150

100



CO

(a)

x 50

2 3 Time (Orbits)


108




E

1 2 Time (Orbits)


2E-5

1 2 Time (Orbits)


OEOft

1 2 3 Time (Orbits)


1 2 3 Time (Orbits)


1 2 Time (Orbits)


1E-5 E z

2

0E0



109















are carried out











110












efficient












111






91 remains constant








by Ms

1 gives

qr = -Ms 1Csqr - Ms

lKsqr



Pu t t ing

v =

+

-M~lCs -M~Cd -1

0

-Mg~lMd

Id

0 v + -M~lKs -M~lKd

0 0

-M-lfs

0



(518)

(519)

x =Ax + Bud + Fd (520)

112

where

A = MC MK Id12 o

24x24

and

Let

B = ^ U 5 R 2 4 X 3 -

Fd = I FJ ) e f t 2 4 1

mdash _ i _ -5

(521)

(522)

(523)

(524)







S=Ax + Bud (526)






K(o)sx

V

dx ~ dXQ dy ~ dyQ

d z - d z0 J

Jy

dx da XQ vo

d z - dZQ ) dy dyQ

(527)

113


























114

u

MODEL UNCERTAINTY

SYSTEM PLANT

y

u LQR CONTROLLER

X KBF ESTIMATOR

u

y

LQG COMPENSATOR

PLANT

X = f v ( X t )

COMPENSATOR

x = A k x - B k y

ud

= C k f

y


115














summarized below



(528)


0 0 $u(lt) 0 0 0 0 1

0 0 1 0 0 0 0 1

(529)


components of xu


Qu =

bull1 0 0 o -0 1 0 0 0 0 1 0

0 0 0 10

(530)

116


selected as 1 0 0 0 0 1 0 0 0 0 4 0

LO 0 0 1


(531)

i o 0 15

(532)





where r is a scaler













117

100

co

gt CO

-50

-100 10


Target r = 5x10 5

r = 0 (LQG)


(a)


101 10


100

50

X3

gt 0

CO

-50

bull100 10




(b)


101 10


118


parts[63]



Dv = Cvxv


(534)

Cy

0 0 0 0 (0) 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0

h 0 0 0

0 0 0 lt ( 0 ) o 0

0 1 0 0 0 1

0 0 0

2 r

h 0 0

0 0 1 0 0 1

(535)


O f Xy


Qv

10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

(536)

mdash


1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 x 10 4 0

0 0 0 0 0 0 0 9 x 10 4

(537)

119


1 0 0 0 0 1 0 0 0 0 5 x 10 4 0

LO 0 0 5 x 10 4

(538)





(539)







these conditions



xf =

$d =

Aku

0

0 Akv

Cku o

Cu 0

Bku

0 B Xf

Xf = Ckxf

0

kv

Hf = Akxf - Bkyf

(540)

Vu Vv 0 Cv

Xf = CfXf



120


X

(541)

Ckxf















vibration case




tems

121






122





t o

1 2 Time (Orbits)



(b)

Figure 5-8



123

























124


























125











angles

















126

flexible tether






















based experiments


dynamics

127

BIBLIOGRAPHY
















128















129














130

6 1995 pp 373-384




















Letter V o l 2 No 1 1982 pp 48-52


131








132























133





w = Av (12)


4-1 ltL3gt j







equation Similarly

E = aAB + bCD (15)








notation

Wi = AjiVj (17)

134




tensor notation as

di-dx-j-1 ( L 8 )


f - a )


is given by

C = Ax)Bx) + A(x)Bx) (110)






13 Forcing Function








135


Mij = qsRv

ni^MtnrnRv

mj + Rv

niMtnTn^Rv

mj + Rv

niMtnmRv

mjs) (113)


-JPdMu 1 bdquo T x






dq dq dqt = dq^QP^




Fk(q^q^)=(RP

sk + R P n J q n ) ^ - - Q d k





(118)

136

















attitude angles

ci = on + 6

Pi = Pi (HI)

7t = 0



137



-S 0 c Pi

(II2)



(215)



(II3) Vi = I Pi i


where

and



(II4)

(II5)

(II6)

D1 = D D l D S v

52 = 31 + P(d2)m +T[d2

D3 = D2 + l2 + dX2Pii)m + T2i6x






138


KH = miDD + 2DiP






TV



d t dqred d(ired

with





139























v





A P P E N D I C E S








v i

LIST OF SYMBOLS





respectively








true anomaly




tion








m

Pi

w 0 i

v i i






modulus E






K A Z $i(k + dXi+1)











0N) order-N




matrix

v i i i








th



respectively





in the ith link






F-Q mdash








ix




- $ T



qv---QtNT


adjacent links)



i + $DJi




tions respectively





LIST OF FIGURES


satellite system 2

























xi













without offset










without offset






x i i






















the local vertical








x i i i








xiv



















xv

1 INTRODUCTION























1


2

Satellite N


2


3



























4





tether





















5


lt mdash

lt Satellite


6


























7















ML+F = Q





be considered






8




























9



























10



























11

two methods[39]
























12





motion



modes










( L Q G L T - R )








13


21 Kinematics















frame



Di + Ti(i + f(i)) (21)

14

15


Ui nvi

wi I I i = 1

nzi bull

(22)



taken as 2 j - l

Hi(xiii)=[j) (23)




k (24)





ff(i) = $i(xili)5i(t) (25)


0 0

^i(xiJi) =


0

0 0

0

G 5R 3 x n f

(26)



16






( 3 d

dx~ + dT)^Xili^ ( 2 8 )

one arrives at











1 P i tch F[ = Cf ( a j )F 0



17

where

c f (A) =

-Sai Cai

0 0

and

C(7i ) =

0 -s0i] = 0 1 0

0 cpi

i 0 0 = 0

0

(211)

(212)

(213)








(214) SliCPi CliCPi









18



















(219)

that

(220)

19


20


cos(0i) 0 0 s i n ( 0 i ) 0 0

0 0 1

Dri

h (221)

= DTlDSv




1 1 1 1 J (222)

= D D l D 3 v


(223)



0 0 1

DTl = D D i = I d 3 i = 2N (224)


2N (225)








21





(226)





and

Ai = i(ti) - ^(t 0 -) = - ^ ( t 2 i ) (227)


V = lUli l^lL (228)

1 tf-tQ v y

A t = (229)


221 Kinetic energy




4 ^ ^ PM) T i $ 4 TiSi]i[t (231)

22

where

m

k J

x l (232)



rewritten as

1 T

(233)


Mt =



pound Sfttiqxnq

(234)




i=l (235)



(236) Mt =

0 0

0 Mt2

0

0 0

MH

0 0 0

0 0 0 bull MtN



23


Mti

m D D l


sym 0

0


0 Id nfj

0 m~

i = 1 3 5 N

(237)


gets

D D l

1 D D l = 1 0 0 0 D n

2 0 (238) 0 0 1










sym sym



tr(ffTai

TTaifi) tr(ffTai





(241)



24

and

j tr(A) =tr(^J A^j (243)



P(fi)Pii)dmi =

tr(Tai

TTaiImi) tTTai


TT7Jmi) sym sym sym

(244)


expanded is


sym sym


sym J z^dmi J


(245)






can be written as

V9i f dm f


dm

inn Lgti + T j ^ | (246)





(2-lt 7)

25






readily that



224 Strain energy












(249)




26




by





number of modes







EX^Ei+jEi (252)





experimentally [49]

27


jet = (254) wo-





(ii) Jo




dWd

^ = - i p = 2 4 ( 2 5 7 )

= 0 i = 13 N









28

system

Mq t)t+ F(q q t) = Q(q q t) (259)


















Let

e Unqxl (261) m


29











3 = 1



k=l



Qt Jfe=l

where

and


T fc-1 0 TkKAk

0 0 0 0 0 0 0 0 0 0 0 0 -

r T i - i 0 0 0 0 0 0 0 0 0 G

0 0 0 1

Qt = RpQ

G R n lt x l

(265)

(266)

G Unq x l (267)

(268)

30

where

RP

R 1 0 0 R R 2 0 R RP

2 R 3

0 0 0

R RP R PN _ 1 RN J

(269)




transformations



t h l ink as

D = A - l + Ti_iK + + K A i _ i ^ _ i


(271) (











31




t = Rvil (2-74)


Here

Rv


0 0 0

T3V nN-l

R bullN

e Mnqqxl

Ri =


e K n lt x l

RS

T i-1 0 0 o-0 0 0 0 0 0

0 0 0 1

G Mnq x l

(275)

(276)

(277)




(278)


0 0

0 0

0 0

G $lnq x l


(279)

pound = Rnjt + Rdq


32

where

and

- 0 0 0 RI 0 0

Rn = 0 0

0 0

Rf 0 0 0 rgtd K2 0

Rd = 0 0 Rl

0 0 0

RN-1 0

G 5R n w x l

0 0 0 e 5 R N ^ X L

(281)

(282)






where

D D T i


DTL 0 0 0 d 3 0 0 0 ID

0 0 nfj 0

RP =


2 R3

iJfDTTi i^

x l

0 0 0

R PN-I RN

(284)

(285)



33



0 0 0 0 0 0 0 0

(286)









M = RV MtRv


(289)

(290)

Inverting Eq(290)



34


Mr1

h 0 0

0 0

0 0

0 0 0

0

0

0

tN

(292)



operation








Mq + f = P C A (293)




pcTM-pc

= ( dc + PdegTM-^f) (294)




35


















nu ^5 Qk=E Pern ^ (2-97)

m=l deg Q k






36






Defining


= d + dfY + T 3 f c m 3



(298)

(299)



(2100)

as i - 2

k=l i1 cm (2101)



WjRi A R i (2102)




1 Q

Q3

3 -1

T 3 T t 2 (2103)

37

Satellite 3

mdash gt -raquo cm

D1=d1

mdash gt

a mdash gt

o


38




Q

~Qgt1

0

0

( T ^ 2 ) (2104)







(2105)

where





3

Qk = ^ F e i

i=l

d

dqk (2106)




39


40

be written as


QiMmi



(2107)

(2108)



Q

Q T 3 0 Q T 5 0

0 0 0

0 lt33T3 Qyen Q5

3T5 0

0 0 0 Q4T5 0

0 0 0 Q5

5T5

M M I

Ti 2 m 3

Quu (2109)


41










numerically














42















by letting

thus

X - ^ ^ f ^ ) =ltbullbull) (3 -2 )







43








plott ing package



















44

INIT

DGEAR PARAM

IC amp SYS PARAM

MAIN

lt

DGEAR

lt

FCN

OUTPUT



VIBRATION

r OFFS DYNA

ET MICS


45









longer stored




the system











46

h =

1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760



0 400 0 0 0 400












fcm3 = 10 0T










47




OOEOh

-50E-12

-10E-11 I i i



48



(a)

20E6

10E6r-

00E0

-10E6

-20E6


-

1 I I I 1 I I

I I

AP e ~_

i i 1 i i i i

00E0

UJ -50E-12 U J

lt


-10E-11h

(b) Time (Orbits)


49


























50

a p(0) = 2deg 0(0) = 2C

6(0) = 08 m

0(0) = 001 m


D p y = 0 D p 2 = 0

a oo

c -ooo -0 01


51






^MAAAAAAAWVWWWWVVVW

V bull i bull i I i J 00 05 10 15 20


Time (Orbits)


52
























53


1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760

h =

h = 200 0 0

0 400 0 0 0 400















mdash







54



55




























56







57







(b) Time (Orbits)


58










by the coupling

















59








60



Time (Orbits)



Time (Orbits)



0 1 2 3 4 5 (b) Time (Orbits)


61





Tether Pitch Angle


Tether Roll Angle

CD

edT

1 2 3 Time (Orbits)




CD

C O

(a)

Figure 4-




62



0 1 2 3 4 5







0 1 2 3 4 5 (b) Time (Orbits)


63



2 3 4 Time (Orbits)

Tether Pitch Angle


Tether Roll Angle



2 3 4 Time (Orbits)


(a)

Figure 4-


2 3 4 Time (Orbits)


64




(b) Time (Orbits)


65










V0 = 146 ms
















66



i i i i i i i i i i i 0 1 2 3 4 5 0 1 2 3 4 5






67



0 1 2 3 4 5


(b) Time (Orbits)


68




E





Tether Pitch Angle


Tether Roll Angle

CD

oo ax


_ 1 _ L





-50

(a) 1 2 3 4


Time (Orbits)


69




to

-02

(b)

Figure 4-7

2 3 Time (Orbits)


0 1 1 bull 1 1



5

1 1 1 1 1 i i | i i | ^

_ i i i i L -I



70












roll

45 Tether Retrieval










and subsatellite



71




E


1 2 3 Time (Orbits)




Tether Pitch Angle

1 2 3 Time (Orbits)

Tether Roll Angle

i i i i J I I i _





D )

T J ^ 0 E 0 eg

CQ

-3E-3

(a) 1 2 3 4


Time (Orbits)


7 2





n 1 1 1 f

I I I _j I I I I I 1 1 1 1 1 2 3 4 5



Time (Orbits)


(b)

Figure 4-8

73




(a) Time (Orbits)



i i i i i i i i i i I 0 1 2 3






04- -



74




E 10 to


1 2 Time (Orbits)


(b) Time (Orbits)


75

























76


77



o 1 2 Time (Orbits)


0 1 2 Time (Orbits)






78


p1(0)=p2(0)=(33(0)=pt1(0)=pt2(0)=10





79



























80






i i i i i 0 1 2 3 4 5










81






82



























83



I- I i t i i i i I H 0 1 2 0 1 2





84



laquo = 5 E - 3 |

000 002



Time (Orbits)


85




E

2000

cu T3

2000

co CD

2000

C D CD

T J


1 2 Time (Orbits)


cn CD

33 cdl




1 2 Time (Orbits)




86



5E-3h

co

0E0

025

000 E to

-025

002

-002

(b)

Figure 4-14




Time (Orbits)


87






















88





89


1E-2

to

OEO

E 00

to


005



Time (Orbits)


90





91




92

























93


present case













mance











94





by Eq(2109)

Lett ing

~fr = fr~ QuU (52)


ir = s(frMrdc) (53)



mdash = Fr - Qru





Ire = Frc ~~ Qrcu

(56)

mdash vrci






an(^ qrcd

95









-In ) 0 5 V T ^ r (59)


V (510)








qrcd 37Vxl (511)

96


e U 3 N x l (512)

E K 3 7 V x l (513)

















97

and

o 0

Qrcj mdash

o o

0 0

Oiit) 0 0

1

N

N

1deg I o







20

CO CD

T J

cdT

cT

- mdash mdash 1 1 1 1 1 I - 1

Pitch -

A Roll -

A -

1 bull bull

-

1 Time (Orbits)


1 2 Time (Orbits)


gt to N

to


145h

(a) Time (Orbits)


98

1





99
























reduced


100



c n

T J

cdT




c n CD

cdT a

o T J

CQ

- - I 1 1 1 I 1 1 1 1

P i t c h

R o l l

bull

1 -

0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


E gt

10 N

CO


145h

(a) Time (Orbits)


101






102






CD

eg CQ

0 1 2 Time (Orbits)


1 2 Time (Orbits)


gt CO


145h

(a) Time (Orbits)


103





0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


-07 o 1


1E-5

0E0F

Time (Orbits)


104





o 1 2 Time (Orbits)


0 1 2 Time (Orbits)


CD CD

CM CO


to

150

100

1 2 3 0 1 2


to

(a)

50

2 3 Time (Orbits)


105





2 0 h

E 2L


0 5

1 2 3



1 2


1 2 3 Time (Orbits)


2 E - 5

1 2 3


1 2


1 E - 5

E

O E O

Time (Orbits)


106

























107





0 1 2 Time (Orbits)


CO CD

cdT

CO CD

T J CM

CO

1 2 Time (Orbits)



150

100



CO

(a)

x 50

2 3 Time (Orbits)


108




E

1 2 Time (Orbits)


2E-5

1 2 Time (Orbits)


OEOft

1 2 3 Time (Orbits)


1 2 3 Time (Orbits)


1 2 Time (Orbits)


1E-5 E z

2

0E0



109















are carried out











110












efficient












111






91 remains constant








by Ms

1 gives

qr = -Ms 1Csqr - Ms

lKsqr



Pu t t ing

v =

+

-M~lCs -M~Cd -1

0

-Mg~lMd

Id

0 v + -M~lKs -M~lKd

0 0

-M-lfs

0



(518)

(519)

x =Ax + Bud + Fd (520)

112

where

A = MC MK Id12 o

24x24

and

Let

B = ^ U 5 R 2 4 X 3 -

Fd = I FJ ) e f t 2 4 1

mdash _ i _ -5

(521)

(522)

(523)

(524)







S=Ax + Bud (526)






K(o)sx

V

dx ~ dXQ dy ~ dyQ

d z - d z0 J

Jy

dx da XQ vo

d z - dZQ ) dy dyQ

(527)

113


























114

u

MODEL UNCERTAINTY

SYSTEM PLANT

y

u LQR CONTROLLER

X KBF ESTIMATOR

u

y

LQG COMPENSATOR

PLANT

X = f v ( X t )

COMPENSATOR

x = A k x - B k y

ud

= C k f

y


115














summarized below



(528)


0 0 $u(lt) 0 0 0 0 1

0 0 1 0 0 0 0 1

(529)


components of xu


Qu =

bull1 0 0 o -0 1 0 0 0 0 1 0

0 0 0 10

(530)

116


selected as 1 0 0 0 0 1 0 0 0 0 4 0

LO 0 0 1


(531)

i o 0 15

(532)





where r is a scaler













117

100

co

gt CO

-50

-100 10


Target r = 5x10 5

r = 0 (LQG)


(a)


101 10


100

50

X3

gt 0

CO

-50

bull100 10




(b)


101 10


118


parts[63]



Dv = Cvxv


(534)

Cy

0 0 0 0 (0) 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0

h 0 0 0

0 0 0 lt ( 0 ) o 0

0 1 0 0 0 1

0 0 0

2 r

h 0 0

0 0 1 0 0 1

(535)


O f Xy


Qv

10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

(536)

mdash


1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 x 10 4 0

0 0 0 0 0 0 0 9 x 10 4

(537)

119


1 0 0 0 0 1 0 0 0 0 5 x 10 4 0

LO 0 0 5 x 10 4

(538)





(539)







these conditions



xf =

$d =

Aku

0

0 Akv

Cku o

Cu 0

Bku

0 B Xf

Xf = Ckxf

0

kv

Hf = Akxf - Bkyf

(540)

Vu Vv 0 Cv

Xf = CfXf



120


X

(541)

Ckxf















vibration case




tems

121






122





t o

1 2 Time (Orbits)



(b)

Figure 5-8



123

























124


























125











angles

















126

flexible tether






















based experiments


dynamics

127

BIBLIOGRAPHY
















128















129














130

6 1995 pp 373-384




















Letter V o l 2 No 1 1982 pp 48-52


131








132























133





w = Av (12)


4-1 ltL3gt j







equation Similarly

E = aAB + bCD (15)








notation

Wi = AjiVj (17)

134




tensor notation as

di-dx-j-1 ( L 8 )


f - a )


is given by

C = Ax)Bx) + A(x)Bx) (110)






13 Forcing Function








135


Mij = qsRv

ni^MtnrnRv

mj + Rv

niMtnTn^Rv

mj + Rv

niMtnmRv

mjs) (113)


-JPdMu 1 bdquo T x






dq dq dqt = dq^QP^




Fk(q^q^)=(RP

sk + R P n J q n ) ^ - - Q d k





(118)

136

















attitude angles

ci = on + 6

Pi = Pi (HI)

7t = 0



137



-S 0 c Pi

(II2)



(215)



(II3) Vi = I Pi i


where

and



(II4)

(II5)

(II6)

D1 = D D l D S v

52 = 31 + P(d2)m +T[d2

D3 = D2 + l2 + dX2Pii)m + T2i6x






138


KH = miDD + 2DiP






TV



d t dqred d(ired

with





139





A P P E N D I C E S








v i

LIST OF SYMBOLS





respectively








true anomaly




tion








m

Pi

w 0 i

v i i






modulus E






K A Z $i(k + dXi+1)











0N) order-N




matrix

v i i i








th



respectively





in the ith link






F-Q mdash








ix




- $ T



qv---QtNT


adjacent links)



i + $DJi




tions respectively





LIST OF FIGURES


satellite system 2

























xi













without offset










without offset






x i i






















the local vertical








x i i i








xiv



















xv

1 INTRODUCTION























1


2

Satellite N


2


3



























4





tether





















5


lt mdash

lt Satellite


6


























7















ML+F = Q





be considered






8




























9



























10



























11

two methods[39]
























12





motion



modes










( L Q G L T - R )








13


21 Kinematics















frame



Di + Ti(i + f(i)) (21)

14

15


Ui nvi

wi I I i = 1

nzi bull

(22)



taken as 2 j - l

Hi(xiii)=[j) (23)




k (24)





ff(i) = $i(xili)5i(t) (25)


0 0

^i(xiJi) =


0

0 0

0

G 5R 3 x n f

(26)



16






( 3 d

dx~ + dT)^Xili^ ( 2 8 )

one arrives at











1 P i tch F[ = Cf ( a j )F 0



17

where

c f (A) =

-Sai Cai

0 0

and

C(7i ) =

0 -s0i] = 0 1 0

0 cpi

i 0 0 = 0

0

(211)

(212)

(213)








(214) SliCPi CliCPi









18



















(219)

that

(220)

19


20


cos(0i) 0 0 s i n ( 0 i ) 0 0

0 0 1

Dri

h (221)

= DTlDSv




1 1 1 1 J (222)

= D D l D 3 v


(223)



0 0 1

DTl = D D i = I d 3 i = 2N (224)


2N (225)








21





(226)





and

Ai = i(ti) - ^(t 0 -) = - ^ ( t 2 i ) (227)


V = lUli l^lL (228)

1 tf-tQ v y

A t = (229)


221 Kinetic energy




4 ^ ^ PM) T i $ 4 TiSi]i[t (231)

22

where

m

k J

x l (232)



rewritten as

1 T

(233)


Mt =



pound Sfttiqxnq

(234)




i=l (235)



(236) Mt =

0 0

0 Mt2

0

0 0

MH

0 0 0

0 0 0 bull MtN



23


Mti

m D D l


sym 0

0


0 Id nfj

0 m~

i = 1 3 5 N

(237)


gets

D D l

1 D D l = 1 0 0 0 D n

2 0 (238) 0 0 1










sym sym



tr(ffTai

TTaifi) tr(ffTai





(241)



24

and

j tr(A) =tr(^J A^j (243)



P(fi)Pii)dmi =

tr(Tai

TTaiImi) tTTai


TT7Jmi) sym sym sym

(244)


expanded is


sym sym


sym J z^dmi J


(245)






can be written as

V9i f dm f


dm

inn Lgti + T j ^ | (246)





(2-lt 7)

25






readily that



224 Strain energy












(249)




26




by





number of modes







EX^Ei+jEi (252)





experimentally [49]

27


jet = (254) wo-





(ii) Jo




dWd

^ = - i p = 2 4 ( 2 5 7 )

= 0 i = 13 N









28

system

Mq t)t+ F(q q t) = Q(q q t) (259)


















Let

e Unqxl (261) m


29











3 = 1



k=l



Qt Jfe=l

where

and


T fc-1 0 TkKAk

0 0 0 0 0 0 0 0 0 0 0 0 -

r T i - i 0 0 0 0 0 0 0 0 0 G

0 0 0 1

Qt = RpQ

G R n lt x l

(265)

(266)

G Unq x l (267)

(268)

30

where

RP

R 1 0 0 R R 2 0 R RP

2 R 3

0 0 0

R RP R PN _ 1 RN J

(269)




transformations



t h l ink as

D = A - l + Ti_iK + + K A i _ i ^ _ i


(271) (











31




t = Rvil (2-74)


Here

Rv


0 0 0

T3V nN-l

R bullN

e Mnqqxl

Ri =


e K n lt x l

RS

T i-1 0 0 o-0 0 0 0 0 0

0 0 0 1

G Mnq x l

(275)

(276)

(277)




(278)


0 0

0 0

0 0

G $lnq x l


(279)

pound = Rnjt + Rdq


32

where

and

- 0 0 0 RI 0 0

Rn = 0 0

0 0

Rf 0 0 0 rgtd K2 0

Rd = 0 0 Rl

0 0 0

RN-1 0

G 5R n w x l

0 0 0 e 5 R N ^ X L

(281)

(282)






where

D D T i


DTL 0 0 0 d 3 0 0 0 ID

0 0 nfj 0

RP =


2 R3

iJfDTTi i^

x l

0 0 0

R PN-I RN

(284)

(285)



33



0 0 0 0 0 0 0 0

(286)









M = RV MtRv


(289)

(290)

Inverting Eq(290)



34


Mr1

h 0 0

0 0

0 0

0 0 0

0

0

0

tN

(292)



operation








Mq + f = P C A (293)




pcTM-pc

= ( dc + PdegTM-^f) (294)




35


















nu ^5 Qk=E Pern ^ (2-97)

m=l deg Q k






36






Defining


= d + dfY + T 3 f c m 3



(298)

(299)



(2100)

as i - 2

k=l i1 cm (2101)



WjRi A R i (2102)




1 Q

Q3

3 -1

T 3 T t 2 (2103)

37

Satellite 3

mdash gt -raquo cm

D1=d1

mdash gt

a mdash gt

o


38




Q

~Qgt1

0

0

( T ^ 2 ) (2104)







(2105)

where





3

Qk = ^ F e i

i=l

d

dqk (2106)




39


40

be written as


QiMmi



(2107)

(2108)



Q

Q T 3 0 Q T 5 0

0 0 0

0 lt33T3 Qyen Q5

3T5 0

0 0 0 Q4T5 0

0 0 0 Q5

5T5

M M I

Ti 2 m 3

Quu (2109)


41










numerically














42















by letting

thus

X - ^ ^ f ^ ) =ltbullbull) (3 -2 )







43








plott ing package



















44

INIT

DGEAR PARAM

IC amp SYS PARAM

MAIN

lt

DGEAR

lt

FCN

OUTPUT



VIBRATION

r OFFS DYNA

ET MICS


45









longer stored




the system











46

h =

1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760



0 400 0 0 0 400












fcm3 = 10 0T










47




OOEOh

-50E-12

-10E-11 I i i



48



(a)

20E6

10E6r-

00E0

-10E6

-20E6


-

1 I I I 1 I I

I I

AP e ~_

i i 1 i i i i

00E0

UJ -50E-12 U J

lt


-10E-11h

(b) Time (Orbits)


49


























50

a p(0) = 2deg 0(0) = 2C

6(0) = 08 m

0(0) = 001 m


D p y = 0 D p 2 = 0

a oo

c -ooo -0 01


51






^MAAAAAAAWVWWWWVVVW

V bull i bull i I i J 00 05 10 15 20


Time (Orbits)


52
























53


1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760

h =

h = 200 0 0

0 400 0 0 0 400















mdash







54



55




























56







57







(b) Time (Orbits)


58










by the coupling

















59








60



Time (Orbits)



Time (Orbits)



0 1 2 3 4 5 (b) Time (Orbits)


61





Tether Pitch Angle


Tether Roll Angle

CD

edT

1 2 3 Time (Orbits)




CD

C O

(a)

Figure 4-




62



0 1 2 3 4 5







0 1 2 3 4 5 (b) Time (Orbits)


63



2 3 4 Time (Orbits)

Tether Pitch Angle


Tether Roll Angle



2 3 4 Time (Orbits)


(a)

Figure 4-


2 3 4 Time (Orbits)


64




(b) Time (Orbits)


65










V0 = 146 ms
















66



i i i i i i i i i i i 0 1 2 3 4 5 0 1 2 3 4 5






67



0 1 2 3 4 5


(b) Time (Orbits)


68




E





Tether Pitch Angle


Tether Roll Angle

CD

oo ax


_ 1 _ L





-50

(a) 1 2 3 4


Time (Orbits)


69




to

-02

(b)

Figure 4-7

2 3 Time (Orbits)


0 1 1 bull 1 1



5

1 1 1 1 1 i i | i i | ^

_ i i i i L -I



70












roll

45 Tether Retrieval










and subsatellite



71




E


1 2 3 Time (Orbits)




Tether Pitch Angle

1 2 3 Time (Orbits)

Tether Roll Angle

i i i i J I I i _





D )

T J ^ 0 E 0 eg

CQ

-3E-3

(a) 1 2 3 4


Time (Orbits)


7 2





n 1 1 1 f

I I I _j I I I I I 1 1 1 1 1 2 3 4 5



Time (Orbits)


(b)

Figure 4-8

73




(a) Time (Orbits)



i i i i i i i i i i I 0 1 2 3






04- -



74




E 10 to


1 2 Time (Orbits)


(b) Time (Orbits)


75

























76


77



o 1 2 Time (Orbits)


0 1 2 Time (Orbits)






78


p1(0)=p2(0)=(33(0)=pt1(0)=pt2(0)=10





79



























80






i i i i i 0 1 2 3 4 5










81






82



























83



I- I i t i i i i I H 0 1 2 0 1 2





84



laquo = 5 E - 3 |

000 002



Time (Orbits)


85




E

2000

cu T3

2000

co CD

2000

C D CD

T J


1 2 Time (Orbits)


cn CD

33 cdl




1 2 Time (Orbits)




86



5E-3h

co

0E0

025

000 E to

-025

002

-002

(b)

Figure 4-14




Time (Orbits)


87






















88





89


1E-2

to

OEO

E 00

to


005



Time (Orbits)


90





91




92

























93


present case













mance











94





by Eq(2109)

Lett ing

~fr = fr~ QuU (52)


ir = s(frMrdc) (53)



mdash = Fr - Qru





Ire = Frc ~~ Qrcu

(56)

mdash vrci






an(^ qrcd

95









-In ) 0 5 V T ^ r (59)


V (510)








qrcd 37Vxl (511)

96


e U 3 N x l (512)

E K 3 7 V x l (513)

















97

and

o 0

Qrcj mdash

o o

0 0

Oiit) 0 0

1

N

N

1deg I o







20

CO CD

T J

cdT

cT

- mdash mdash 1 1 1 1 1 I - 1

Pitch -

A Roll -

A -

1 bull bull

-

1 Time (Orbits)


1 2 Time (Orbits)


gt to N

to


145h

(a) Time (Orbits)


98

1





99
























reduced


100



c n

T J

cdT




c n CD

cdT a

o T J

CQ

- - I 1 1 1 I 1 1 1 1

P i t c h

R o l l

bull

1 -

0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


E gt

10 N

CO


145h

(a) Time (Orbits)


101






102






CD

eg CQ

0 1 2 Time (Orbits)


1 2 Time (Orbits)


gt CO


145h

(a) Time (Orbits)


103





0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


-07 o 1


1E-5

0E0F

Time (Orbits)


104





o 1 2 Time (Orbits)


0 1 2 Time (Orbits)


CD CD

CM CO


to

150

100

1 2 3 0 1 2


to

(a)

50

2 3 Time (Orbits)


105





2 0 h

E 2L


0 5

1 2 3



1 2


1 2 3 Time (Orbits)


2 E - 5

1 2 3


1 2


1 E - 5

E

O E O

Time (Orbits)


106

























107





0 1 2 Time (Orbits)


CO CD

cdT

CO CD

T J CM

CO

1 2 Time (Orbits)



150

100



CO

(a)

x 50

2 3 Time (Orbits)


108




E

1 2 Time (Orbits)


2E-5

1 2 Time (Orbits)


OEOft

1 2 3 Time (Orbits)


1 2 3 Time (Orbits)


1 2 Time (Orbits)


1E-5 E z

2

0E0



109















are carried out











110












efficient












111






91 remains constant








by Ms

1 gives

qr = -Ms 1Csqr - Ms

lKsqr



Pu t t ing

v =

+

-M~lCs -M~Cd -1

0

-Mg~lMd

Id

0 v + -M~lKs -M~lKd

0 0

-M-lfs

0



(518)

(519)

x =Ax + Bud + Fd (520)

112

where

A = MC MK Id12 o

24x24

and

Let

B = ^ U 5 R 2 4 X 3 -

Fd = I FJ ) e f t 2 4 1

mdash _ i _ -5

(521)

(522)

(523)

(524)







S=Ax + Bud (526)






K(o)sx

V

dx ~ dXQ dy ~ dyQ

d z - d z0 J

Jy

dx da XQ vo

d z - dZQ ) dy dyQ

(527)

113


























114

u

MODEL UNCERTAINTY

SYSTEM PLANT

y

u LQR CONTROLLER

X KBF ESTIMATOR

u

y

LQG COMPENSATOR

PLANT

X = f v ( X t )

COMPENSATOR

x = A k x - B k y

ud

= C k f

y


115














summarized below



(528)


0 0 $u(lt) 0 0 0 0 1

0 0 1 0 0 0 0 1

(529)


components of xu


Qu =

bull1 0 0 o -0 1 0 0 0 0 1 0

0 0 0 10

(530)

116


selected as 1 0 0 0 0 1 0 0 0 0 4 0

LO 0 0 1


(531)

i o 0 15

(532)





where r is a scaler













117

100

co

gt CO

-50

-100 10


Target r = 5x10 5

r = 0 (LQG)


(a)


101 10


100

50

X3

gt 0

CO

-50

bull100 10




(b)


101 10


118


parts[63]



Dv = Cvxv


(534)

Cy

0 0 0 0 (0) 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0

h 0 0 0

0 0 0 lt ( 0 ) o 0

0 1 0 0 0 1

0 0 0

2 r

h 0 0

0 0 1 0 0 1

(535)


O f Xy


Qv

10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

(536)

mdash


1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 x 10 4 0

0 0 0 0 0 0 0 9 x 10 4

(537)

119


1 0 0 0 0 1 0 0 0 0 5 x 10 4 0

LO 0 0 5 x 10 4

(538)





(539)







these conditions



xf =

$d =

Aku

0

0 Akv

Cku o

Cu 0

Bku

0 B Xf

Xf = Ckxf

0

kv

Hf = Akxf - Bkyf

(540)

Vu Vv 0 Cv

Xf = CfXf



120


X

(541)

Ckxf















vibration case




tems

121






122





t o

1 2 Time (Orbits)



(b)

Figure 5-8



123

























124


























125











angles

















126

flexible tether






















based experiments


dynamics

127

BIBLIOGRAPHY
















128















129














130

6 1995 pp 373-384




















Letter V o l 2 No 1 1982 pp 48-52


131








132























133





w = Av (12)


4-1 ltL3gt j







equation Similarly

E = aAB + bCD (15)








notation

Wi = AjiVj (17)

134




tensor notation as

di-dx-j-1 ( L 8 )


f - a )


is given by

C = Ax)Bx) + A(x)Bx) (110)






13 Forcing Function








135


Mij = qsRv

ni^MtnrnRv

mj + Rv

niMtnTn^Rv

mj + Rv

niMtnmRv

mjs) (113)


-JPdMu 1 bdquo T x






dq dq dqt = dq^QP^




Fk(q^q^)=(RP

sk + R P n J q n ) ^ - - Q d k





(118)

136

















attitude angles

ci = on + 6

Pi = Pi (HI)

7t = 0



137



-S 0 c Pi

(II2)



(215)



(II3) Vi = I Pi i


where

and



(II4)

(II5)

(II6)

D1 = D D l D S v

52 = 31 + P(d2)m +T[d2

D3 = D2 + l2 + dX2Pii)m + T2i6x






138


KH = miDD + 2DiP






TV



d t dqred d(ired

with





139

LIST OF SYMBOLS





respectively








true anomaly




tion








m

Pi

w 0 i

v i i






modulus E






K A Z $i(k + dXi+1)











0N) order-N




matrix

v i i i








th



respectively





in the ith link






F-Q mdash








ix




- $ T



qv---QtNT


adjacent links)



i + $DJi




tions respectively





LIST OF FIGURES


satellite system 2

























xi













without offset










without offset






x i i






















the local vertical








x i i i








xiv



















xv

1 INTRODUCTION























1


2

Satellite N


2


3



























4





tether





















5


lt mdash

lt Satellite


6


























7















ML+F = Q





be considered






8




























9



























10



























11

two methods[39]
























12





motion



modes










( L Q G L T - R )








13


21 Kinematics















frame



Di + Ti(i + f(i)) (21)

14

15


Ui nvi

wi I I i = 1

nzi bull

(22)



taken as 2 j - l

Hi(xiii)=[j) (23)




k (24)





ff(i) = $i(xili)5i(t) (25)


0 0

^i(xiJi) =


0

0 0

0

G 5R 3 x n f

(26)



16






( 3 d

dx~ + dT)^Xili^ ( 2 8 )

one arrives at











1 P i tch F[ = Cf ( a j )F 0



17

where

c f (A) =

-Sai Cai

0 0

and

C(7i ) =

0 -s0i] = 0 1 0

0 cpi

i 0 0 = 0

0

(211)

(212)

(213)








(214) SliCPi CliCPi









18



















(219)

that

(220)

19


20


cos(0i) 0 0 s i n ( 0 i ) 0 0

0 0 1

Dri

h (221)

= DTlDSv




1 1 1 1 J (222)

= D D l D 3 v


(223)



0 0 1

DTl = D D i = I d 3 i = 2N (224)


2N (225)








21





(226)





and

Ai = i(ti) - ^(t 0 -) = - ^ ( t 2 i ) (227)


V = lUli l^lL (228)

1 tf-tQ v y

A t = (229)


221 Kinetic energy




4 ^ ^ PM) T i $ 4 TiSi]i[t (231)

22

where

m

k J

x l (232)



rewritten as

1 T

(233)


Mt =



pound Sfttiqxnq

(234)




i=l (235)



(236) Mt =

0 0

0 Mt2

0

0 0

MH

0 0 0

0 0 0 bull MtN



23


Mti

m D D l


sym 0

0


0 Id nfj

0 m~

i = 1 3 5 N

(237)


gets

D D l

1 D D l = 1 0 0 0 D n

2 0 (238) 0 0 1










sym sym



tr(ffTai

TTaifi) tr(ffTai





(241)



24

and

j tr(A) =tr(^J A^j (243)



P(fi)Pii)dmi =

tr(Tai

TTaiImi) tTTai


TT7Jmi) sym sym sym

(244)


expanded is


sym sym


sym J z^dmi J


(245)






can be written as

V9i f dm f


dm

inn Lgti + T j ^ | (246)





(2-lt 7)

25






readily that



224 Strain energy












(249)




26




by





number of modes







EX^Ei+jEi (252)





experimentally [49]

27


jet = (254) wo-





(ii) Jo




dWd

^ = - i p = 2 4 ( 2 5 7 )

= 0 i = 13 N









28

system

Mq t)t+ F(q q t) = Q(q q t) (259)


















Let

e Unqxl (261) m


29











3 = 1



k=l



Qt Jfe=l

where

and


T fc-1 0 TkKAk

0 0 0 0 0 0 0 0 0 0 0 0 -

r T i - i 0 0 0 0 0 0 0 0 0 G

0 0 0 1

Qt = RpQ

G R n lt x l

(265)

(266)

G Unq x l (267)

(268)

30

where

RP

R 1 0 0 R R 2 0 R RP

2 R 3

0 0 0

R RP R PN _ 1 RN J

(269)




transformations



t h l ink as

D = A - l + Ti_iK + + K A i _ i ^ _ i


(271) (











31




t = Rvil (2-74)


Here

Rv


0 0 0

T3V nN-l

R bullN

e Mnqqxl

Ri =


e K n lt x l

RS

T i-1 0 0 o-0 0 0 0 0 0

0 0 0 1

G Mnq x l

(275)

(276)

(277)




(278)


0 0

0 0

0 0

G $lnq x l


(279)

pound = Rnjt + Rdq


32

where

and

- 0 0 0 RI 0 0

Rn = 0 0

0 0

Rf 0 0 0 rgtd K2 0

Rd = 0 0 Rl

0 0 0

RN-1 0

G 5R n w x l

0 0 0 e 5 R N ^ X L

(281)

(282)






where

D D T i


DTL 0 0 0 d 3 0 0 0 ID

0 0 nfj 0

RP =


2 R3

iJfDTTi i^

x l

0 0 0

R PN-I RN

(284)

(285)



33



0 0 0 0 0 0 0 0

(286)









M = RV MtRv


(289)

(290)

Inverting Eq(290)



34


Mr1

h 0 0

0 0

0 0

0 0 0

0

0

0

tN

(292)



operation








Mq + f = P C A (293)




pcTM-pc

= ( dc + PdegTM-^f) (294)




35


















nu ^5 Qk=E Pern ^ (2-97)

m=l deg Q k






36






Defining


= d + dfY + T 3 f c m 3



(298)

(299)



(2100)

as i - 2

k=l i1 cm (2101)



WjRi A R i (2102)




1 Q

Q3

3 -1

T 3 T t 2 (2103)

37

Satellite 3

mdash gt -raquo cm

D1=d1

mdash gt

a mdash gt

o


38




Q

~Qgt1

0

0

( T ^ 2 ) (2104)







(2105)

where





3

Qk = ^ F e i

i=l

d

dqk (2106)




39


40

be written as


QiMmi



(2107)

(2108)



Q

Q T 3 0 Q T 5 0

0 0 0

0 lt33T3 Qyen Q5

3T5 0

0 0 0 Q4T5 0

0 0 0 Q5

5T5

M M I

Ti 2 m 3

Quu (2109)


41










numerically














42















by letting

thus

X - ^ ^ f ^ ) =ltbullbull) (3 -2 )







43








plott ing package



















44

INIT

DGEAR PARAM

IC amp SYS PARAM

MAIN

lt

DGEAR

lt

FCN

OUTPUT



VIBRATION

r OFFS DYNA

ET MICS


45









longer stored




the system











46

h =

1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760



0 400 0 0 0 400












fcm3 = 10 0T










47




OOEOh

-50E-12

-10E-11 I i i



48



(a)

20E6

10E6r-

00E0

-10E6

-20E6


-

1 I I I 1 I I

I I

AP e ~_

i i 1 i i i i

00E0

UJ -50E-12 U J

lt


-10E-11h

(b) Time (Orbits)


49


























50

a p(0) = 2deg 0(0) = 2C

6(0) = 08 m

0(0) = 001 m


D p y = 0 D p 2 = 0

a oo

c -ooo -0 01


51






^MAAAAAAAWVWWWWVVVW

V bull i bull i I i J 00 05 10 15 20


Time (Orbits)


52
























53


1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760

h =

h = 200 0 0

0 400 0 0 0 400















mdash







54



55




























56







57







(b) Time (Orbits)


58










by the coupling

















59








60



Time (Orbits)



Time (Orbits)



0 1 2 3 4 5 (b) Time (Orbits)


61





Tether Pitch Angle


Tether Roll Angle

CD

edT

1 2 3 Time (Orbits)




CD

C O

(a)

Figure 4-




62



0 1 2 3 4 5







0 1 2 3 4 5 (b) Time (Orbits)


63



2 3 4 Time (Orbits)

Tether Pitch Angle


Tether Roll Angle



2 3 4 Time (Orbits)


(a)

Figure 4-


2 3 4 Time (Orbits)


64




(b) Time (Orbits)


65










V0 = 146 ms
















66



i i i i i i i i i i i 0 1 2 3 4 5 0 1 2 3 4 5






67



0 1 2 3 4 5


(b) Time (Orbits)


68




E





Tether Pitch Angle


Tether Roll Angle

CD

oo ax


_ 1 _ L





-50

(a) 1 2 3 4


Time (Orbits)


69




to

-02

(b)

Figure 4-7

2 3 Time (Orbits)


0 1 1 bull 1 1



5

1 1 1 1 1 i i | i i | ^

_ i i i i L -I



70












roll

45 Tether Retrieval










and subsatellite



71




E


1 2 3 Time (Orbits)




Tether Pitch Angle

1 2 3 Time (Orbits)

Tether Roll Angle

i i i i J I I i _





D )

T J ^ 0 E 0 eg

CQ

-3E-3

(a) 1 2 3 4


Time (Orbits)


7 2





n 1 1 1 f

I I I _j I I I I I 1 1 1 1 1 2 3 4 5



Time (Orbits)


(b)

Figure 4-8

73




(a) Time (Orbits)



i i i i i i i i i i I 0 1 2 3






04- -



74




E 10 to


1 2 Time (Orbits)


(b) Time (Orbits)


75

























76


77



o 1 2 Time (Orbits)


0 1 2 Time (Orbits)






78


p1(0)=p2(0)=(33(0)=pt1(0)=pt2(0)=10





79



























80






i i i i i 0 1 2 3 4 5










81






82



























83



I- I i t i i i i I H 0 1 2 0 1 2





84



laquo = 5 E - 3 |

000 002



Time (Orbits)


85




E

2000

cu T3

2000

co CD

2000

C D CD

T J


1 2 Time (Orbits)


cn CD

33 cdl




1 2 Time (Orbits)




86



5E-3h

co

0E0

025

000 E to

-025

002

-002

(b)

Figure 4-14




Time (Orbits)


87






















88





89


1E-2

to

OEO

E 00

to


005



Time (Orbits)


90





91




92

























93


present case













mance











94





by Eq(2109)

Lett ing

~fr = fr~ QuU (52)


ir = s(frMrdc) (53)



mdash = Fr - Qru





Ire = Frc ~~ Qrcu

(56)

mdash vrci






an(^ qrcd

95









-In ) 0 5 V T ^ r (59)


V (510)








qrcd 37Vxl (511)

96


e U 3 N x l (512)

E K 3 7 V x l (513)

















97

and

o 0

Qrcj mdash

o o

0 0

Oiit) 0 0

1

N

N

1deg I o







20

CO CD

T J

cdT

cT

- mdash mdash 1 1 1 1 1 I - 1

Pitch -

A Roll -

A -

1 bull bull

-

1 Time (Orbits)


1 2 Time (Orbits)


gt to N

to


145h

(a) Time (Orbits)


98

1





99
























reduced


100



c n

T J

cdT




c n CD

cdT a

o T J

CQ

- - I 1 1 1 I 1 1 1 1

P i t c h

R o l l

bull

1 -

0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


E gt

10 N

CO


145h

(a) Time (Orbits)


101






102






CD

eg CQ

0 1 2 Time (Orbits)


1 2 Time (Orbits)


gt CO


145h

(a) Time (Orbits)


103





0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


-07 o 1


1E-5

0E0F

Time (Orbits)


104





o 1 2 Time (Orbits)


0 1 2 Time (Orbits)


CD CD

CM CO


to

150

100

1 2 3 0 1 2


to

(a)

50

2 3 Time (Orbits)


105





2 0 h

E 2L


0 5

1 2 3



1 2


1 2 3 Time (Orbits)


2 E - 5

1 2 3


1 2


1 E - 5

E

O E O

Time (Orbits)


106

























107





0 1 2 Time (Orbits)


CO CD

cdT

CO CD

T J CM

CO

1 2 Time (Orbits)



150

100



CO

(a)

x 50

2 3 Time (Orbits)


108




E

1 2 Time (Orbits)


2E-5

1 2 Time (Orbits)


OEOft

1 2 3 Time (Orbits)


1 2 3 Time (Orbits)


1 2 Time (Orbits)


1E-5 E z

2

0E0



109















are carried out











110












efficient












111






91 remains constant








by Ms

1 gives

qr = -Ms 1Csqr - Ms

lKsqr



Pu t t ing

v =

+

-M~lCs -M~Cd -1

0

-Mg~lMd

Id

0 v + -M~lKs -M~lKd

0 0

-M-lfs

0



(518)

(519)

x =Ax + Bud + Fd (520)

112

where

A = MC MK Id12 o

24x24

and

Let

B = ^ U 5 R 2 4 X 3 -

Fd = I FJ ) e f t 2 4 1

mdash _ i _ -5

(521)

(522)

(523)

(524)







S=Ax + Bud (526)






K(o)sx

V

dx ~ dXQ dy ~ dyQ

d z - d z0 J

Jy

dx da XQ vo

d z - dZQ ) dy dyQ

(527)

113


























114

u

MODEL UNCERTAINTY

SYSTEM PLANT

y

u LQR CONTROLLER

X KBF ESTIMATOR

u

y

LQG COMPENSATOR

PLANT

X = f v ( X t )

COMPENSATOR

x = A k x - B k y

ud

= C k f

y


115














summarized below



(528)


0 0 $u(lt) 0 0 0 0 1

0 0 1 0 0 0 0 1

(529)


components of xu


Qu =

bull1 0 0 o -0 1 0 0 0 0 1 0

0 0 0 10

(530)

116


selected as 1 0 0 0 0 1 0 0 0 0 4 0

LO 0 0 1


(531)

i o 0 15

(532)





where r is a scaler













117

100

co

gt CO

-50

-100 10


Target r = 5x10 5

r = 0 (LQG)


(a)


101 10


100

50

X3

gt 0

CO

-50

bull100 10




(b)


101 10


118


parts[63]



Dv = Cvxv


(534)

Cy

0 0 0 0 (0) 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0

h 0 0 0

0 0 0 lt ( 0 ) o 0

0 1 0 0 0 1

0 0 0

2 r

h 0 0

0 0 1 0 0 1

(535)


O f Xy


Qv

10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

(536)

mdash


1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 x 10 4 0

0 0 0 0 0 0 0 9 x 10 4

(537)

119


1 0 0 0 0 1 0 0 0 0 5 x 10 4 0

LO 0 0 5 x 10 4

(538)





(539)







these conditions



xf =

$d =

Aku

0

0 Akv

Cku o

Cu 0

Bku

0 B Xf

Xf = Ckxf

0

kv

Hf = Akxf - Bkyf

(540)

Vu Vv 0 Cv

Xf = CfXf



120


X

(541)

Ckxf















vibration case




tems

121






122





t o

1 2 Time (Orbits)



(b)

Figure 5-8



123

























124


























125











angles

















126

flexible tether






















based experiments


dynamics

127

BIBLIOGRAPHY
















128















129














130

6 1995 pp 373-384




















Letter V o l 2 No 1 1982 pp 48-52


131








132























133





w = Av (12)


4-1 ltL3gt j







equation Similarly

E = aAB + bCD (15)








notation

Wi = AjiVj (17)

134




tensor notation as

di-dx-j-1 ( L 8 )


f - a )


is given by

C = Ax)Bx) + A(x)Bx) (110)






13 Forcing Function








135


Mij = qsRv

ni^MtnrnRv

mj + Rv

niMtnTn^Rv

mj + Rv

niMtnmRv

mjs) (113)


-JPdMu 1 bdquo T x






dq dq dqt = dq^QP^




Fk(q^q^)=(RP

sk + R P n J q n ) ^ - - Q d k





(118)

136

















attitude angles

ci = on + 6

Pi = Pi (HI)

7t = 0



137



-S 0 c Pi

(II2)



(215)



(II3) Vi = I Pi i


where

and



(II4)

(II5)

(II6)

D1 = D D l D S v

52 = 31 + P(d2)m +T[d2

D3 = D2 + l2 + dX2Pii)m + T2i6x






138


KH = miDD + 2DiP






TV



d t dqred d(ired

with





139






modulus E






K A Z $i(k + dXi+1)











0N) order-N




matrix

v i i i








th



respectively





in the ith link






F-Q mdash








ix




- $ T



qv---QtNT


adjacent links)



i + $DJi




tions respectively





LIST OF FIGURES


satellite system 2

























xi













without offset










without offset






x i i






















the local vertical








x i i i








xiv



















xv

1 INTRODUCTION























1


2

Satellite N


2


3



























4





tether





















5


lt mdash

lt Satellite


6


























7















ML+F = Q





be considered






8




























9



























10



























11

two methods[39]
























12





motion



modes










( L Q G L T - R )








13


21 Kinematics















frame



Di + Ti(i + f(i)) (21)

14

15


Ui nvi

wi I I i = 1

nzi bull

(22)



taken as 2 j - l

Hi(xiii)=[j) (23)




k (24)





ff(i) = $i(xili)5i(t) (25)


0 0

^i(xiJi) =


0

0 0

0

G 5R 3 x n f

(26)



16






( 3 d

dx~ + dT)^Xili^ ( 2 8 )

one arrives at











1 P i tch F[ = Cf ( a j )F 0



17

where

c f (A) =

-Sai Cai

0 0

and

C(7i ) =

0 -s0i] = 0 1 0

0 cpi

i 0 0 = 0

0

(211)

(212)

(213)








(214) SliCPi CliCPi









18



















(219)

that

(220)

19


20


cos(0i) 0 0 s i n ( 0 i ) 0 0

0 0 1

Dri

h (221)

= DTlDSv




1 1 1 1 J (222)

= D D l D 3 v


(223)



0 0 1

DTl = D D i = I d 3 i = 2N (224)


2N (225)








21





(226)





and

Ai = i(ti) - ^(t 0 -) = - ^ ( t 2 i ) (227)


V = lUli l^lL (228)

1 tf-tQ v y

A t = (229)


221 Kinetic energy




4 ^ ^ PM) T i $ 4 TiSi]i[t (231)

22

where

m

k J

x l (232)



rewritten as

1 T

(233)


Mt =



pound Sfttiqxnq

(234)




i=l (235)



(236) Mt =

0 0

0 Mt2

0

0 0

MH

0 0 0

0 0 0 bull MtN



23


Mti

m D D l


sym 0

0


0 Id nfj

0 m~

i = 1 3 5 N

(237)


gets

D D l

1 D D l = 1 0 0 0 D n

2 0 (238) 0 0 1










sym sym



tr(ffTai

TTaifi) tr(ffTai





(241)



24

and

j tr(A) =tr(^J A^j (243)



P(fi)Pii)dmi =

tr(Tai

TTaiImi) tTTai


TT7Jmi) sym sym sym

(244)


expanded is


sym sym


sym J z^dmi J


(245)






can be written as

V9i f dm f


dm

inn Lgti + T j ^ | (246)





(2-lt 7)

25






readily that



224 Strain energy












(249)




26




by





number of modes







EX^Ei+jEi (252)





experimentally [49]

27


jet = (254) wo-





(ii) Jo




dWd

^ = - i p = 2 4 ( 2 5 7 )

= 0 i = 13 N









28

system

Mq t)t+ F(q q t) = Q(q q t) (259)


















Let

e Unqxl (261) m


29











3 = 1



k=l



Qt Jfe=l

where

and


T fc-1 0 TkKAk

0 0 0 0 0 0 0 0 0 0 0 0 -

r T i - i 0 0 0 0 0 0 0 0 0 G

0 0 0 1

Qt = RpQ

G R n lt x l

(265)

(266)

G Unq x l (267)

(268)

30

where

RP

R 1 0 0 R R 2 0 R RP

2 R 3

0 0 0

R RP R PN _ 1 RN J

(269)




transformations



t h l ink as

D = A - l + Ti_iK + + K A i _ i ^ _ i


(271) (











31




t = Rvil (2-74)


Here

Rv


0 0 0

T3V nN-l

R bullN

e Mnqqxl

Ri =


e K n lt x l

RS

T i-1 0 0 o-0 0 0 0 0 0

0 0 0 1

G Mnq x l

(275)

(276)

(277)




(278)


0 0

0 0

0 0

G $lnq x l


(279)

pound = Rnjt + Rdq


32

where

and

- 0 0 0 RI 0 0

Rn = 0 0

0 0

Rf 0 0 0 rgtd K2 0

Rd = 0 0 Rl

0 0 0

RN-1 0

G 5R n w x l

0 0 0 e 5 R N ^ X L

(281)

(282)






where

D D T i


DTL 0 0 0 d 3 0 0 0 ID

0 0 nfj 0

RP =


2 R3

iJfDTTi i^

x l

0 0 0

R PN-I RN

(284)

(285)



33



0 0 0 0 0 0 0 0

(286)









M = RV MtRv


(289)

(290)

Inverting Eq(290)



34


Mr1

h 0 0

0 0

0 0

0 0 0

0

0

0

tN

(292)



operation








Mq + f = P C A (293)




pcTM-pc

= ( dc + PdegTM-^f) (294)




35


















nu ^5 Qk=E Pern ^ (2-97)

m=l deg Q k






36






Defining


= d + dfY + T 3 f c m 3



(298)

(299)



(2100)

as i - 2

k=l i1 cm (2101)



WjRi A R i (2102)




1 Q

Q3

3 -1

T 3 T t 2 (2103)

37

Satellite 3

mdash gt -raquo cm

D1=d1

mdash gt

a mdash gt

o


38




Q

~Qgt1

0

0

( T ^ 2 ) (2104)







(2105)

where





3

Qk = ^ F e i

i=l

d

dqk (2106)




39


40

be written as


QiMmi



(2107)

(2108)



Q

Q T 3 0 Q T 5 0

0 0 0

0 lt33T3 Qyen Q5

3T5 0

0 0 0 Q4T5 0

0 0 0 Q5

5T5

M M I

Ti 2 m 3

Quu (2109)


41










numerically














42















by letting

thus

X - ^ ^ f ^ ) =ltbullbull) (3 -2 )







43








plott ing package



















44

INIT

DGEAR PARAM

IC amp SYS PARAM

MAIN

lt

DGEAR

lt

FCN

OUTPUT



VIBRATION

r OFFS DYNA

ET MICS


45









longer stored




the system











46

h =

1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760



0 400 0 0 0 400












fcm3 = 10 0T










47




OOEOh

-50E-12

-10E-11 I i i



48



(a)

20E6

10E6r-

00E0

-10E6

-20E6


-

1 I I I 1 I I

I I

AP e ~_

i i 1 i i i i

00E0

UJ -50E-12 U J

lt


-10E-11h

(b) Time (Orbits)


49


























50

a p(0) = 2deg 0(0) = 2C

6(0) = 08 m

0(0) = 001 m


D p y = 0 D p 2 = 0

a oo

c -ooo -0 01


51






^MAAAAAAAWVWWWWVVVW

V bull i bull i I i J 00 05 10 15 20


Time (Orbits)


52
























53


1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760

h =

h = 200 0 0

0 400 0 0 0 400















mdash







54



55




























56







57







(b) Time (Orbits)


58










by the coupling

















59








60



Time (Orbits)



Time (Orbits)



0 1 2 3 4 5 (b) Time (Orbits)


61





Tether Pitch Angle


Tether Roll Angle

CD

edT

1 2 3 Time (Orbits)




CD

C O

(a)

Figure 4-




62



0 1 2 3 4 5







0 1 2 3 4 5 (b) Time (Orbits)


63



2 3 4 Time (Orbits)

Tether Pitch Angle


Tether Roll Angle



2 3 4 Time (Orbits)


(a)

Figure 4-


2 3 4 Time (Orbits)


64




(b) Time (Orbits)


65










V0 = 146 ms
















66



i i i i i i i i i i i 0 1 2 3 4 5 0 1 2 3 4 5






67



0 1 2 3 4 5


(b) Time (Orbits)


68




E





Tether Pitch Angle


Tether Roll Angle

CD

oo ax


_ 1 _ L





-50

(a) 1 2 3 4


Time (Orbits)


69




to

-02

(b)

Figure 4-7

2 3 Time (Orbits)


0 1 1 bull 1 1



5

1 1 1 1 1 i i | i i | ^

_ i i i i L -I



70












roll

45 Tether Retrieval










and subsatellite



71




E


1 2 3 Time (Orbits)




Tether Pitch Angle

1 2 3 Time (Orbits)

Tether Roll Angle

i i i i J I I i _





D )

T J ^ 0 E 0 eg

CQ

-3E-3

(a) 1 2 3 4


Time (Orbits)


7 2





n 1 1 1 f

I I I _j I I I I I 1 1 1 1 1 2 3 4 5



Time (Orbits)


(b)

Figure 4-8

73




(a) Time (Orbits)



i i i i i i i i i i I 0 1 2 3






04- -



74




E 10 to


1 2 Time (Orbits)


(b) Time (Orbits)


75

























76


77



o 1 2 Time (Orbits)


0 1 2 Time (Orbits)






78


p1(0)=p2(0)=(33(0)=pt1(0)=pt2(0)=10





79



























80






i i i i i 0 1 2 3 4 5










81






82



























83



I- I i t i i i i I H 0 1 2 0 1 2





84



laquo = 5 E - 3 |

000 002



Time (Orbits)


85




E

2000

cu T3

2000

co CD

2000

C D CD

T J


1 2 Time (Orbits)


cn CD

33 cdl




1 2 Time (Orbits)




86



5E-3h

co

0E0

025

000 E to

-025

002

-002

(b)

Figure 4-14




Time (Orbits)


87






















88





89


1E-2

to

OEO

E 00

to


005



Time (Orbits)


90





91




92

























93


present case













mance











94





by Eq(2109)

Lett ing

~fr = fr~ QuU (52)


ir = s(frMrdc) (53)



mdash = Fr - Qru





Ire = Frc ~~ Qrcu

(56)

mdash vrci






an(^ qrcd

95









-In ) 0 5 V T ^ r (59)


V (510)








qrcd 37Vxl (511)

96


e U 3 N x l (512)

E K 3 7 V x l (513)

















97

and

o 0

Qrcj mdash

o o

0 0

Oiit) 0 0

1

N

N

1deg I o







20

CO CD

T J

cdT

cT

- mdash mdash 1 1 1 1 1 I - 1

Pitch -

A Roll -

A -

1 bull bull

-

1 Time (Orbits)


1 2 Time (Orbits)


gt to N

to


145h

(a) Time (Orbits)


98

1





99
























reduced


100



c n

T J

cdT




c n CD

cdT a

o T J

CQ

- - I 1 1 1 I 1 1 1 1

P i t c h

R o l l

bull

1 -

0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


E gt

10 N

CO


145h

(a) Time (Orbits)


101






102






CD

eg CQ

0 1 2 Time (Orbits)


1 2 Time (Orbits)


gt CO


145h

(a) Time (Orbits)


103





0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


-07 o 1


1E-5

0E0F

Time (Orbits)


104





o 1 2 Time (Orbits)


0 1 2 Time (Orbits)


CD CD

CM CO


to

150

100

1 2 3 0 1 2


to

(a)

50

2 3 Time (Orbits)


105





2 0 h

E 2L


0 5

1 2 3



1 2


1 2 3 Time (Orbits)


2 E - 5

1 2 3


1 2


1 E - 5

E

O E O

Time (Orbits)


106

























107





0 1 2 Time (Orbits)


CO CD

cdT

CO CD

T J CM

CO

1 2 Time (Orbits)



150

100



CO

(a)

x 50

2 3 Time (Orbits)


108




E

1 2 Time (Orbits)


2E-5

1 2 Time (Orbits)


OEOft

1 2 3 Time (Orbits)


1 2 3 Time (Orbits)


1 2 Time (Orbits)


1E-5 E z

2

0E0



109















are carried out











110












efficient












111






91 remains constant








by Ms

1 gives

qr = -Ms 1Csqr - Ms

lKsqr



Pu t t ing

v =

+

-M~lCs -M~Cd -1

0

-Mg~lMd

Id

0 v + -M~lKs -M~lKd

0 0

-M-lfs

0



(518)

(519)

x =Ax + Bud + Fd (520)

112

where

A = MC MK Id12 o

24x24

and

Let

B = ^ U 5 R 2 4 X 3 -

Fd = I FJ ) e f t 2 4 1

mdash _ i _ -5

(521)

(522)

(523)

(524)







S=Ax + Bud (526)






K(o)sx

V

dx ~ dXQ dy ~ dyQ

d z - d z0 J

Jy

dx da XQ vo

d z - dZQ ) dy dyQ

(527)

113


























114

u

MODEL UNCERTAINTY

SYSTEM PLANT

y

u LQR CONTROLLER

X KBF ESTIMATOR

u

y

LQG COMPENSATOR

PLANT

X = f v ( X t )

COMPENSATOR

x = A k x - B k y

ud

= C k f

y


115














summarized below



(528)


0 0 $u(lt) 0 0 0 0 1

0 0 1 0 0 0 0 1

(529)


components of xu


Qu =

bull1 0 0 o -0 1 0 0 0 0 1 0

0 0 0 10

(530)

116


selected as 1 0 0 0 0 1 0 0 0 0 4 0

LO 0 0 1


(531)

i o 0 15

(532)





where r is a scaler













117

100

co

gt CO

-50

-100 10


Target r = 5x10 5

r = 0 (LQG)


(a)


101 10


100

50

X3

gt 0

CO

-50

bull100 10




(b)


101 10


118


parts[63]



Dv = Cvxv


(534)

Cy

0 0 0 0 (0) 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0

h 0 0 0

0 0 0 lt ( 0 ) o 0

0 1 0 0 0 1

0 0 0

2 r

h 0 0

0 0 1 0 0 1

(535)


O f Xy


Qv

10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

(536)

mdash


1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 x 10 4 0

0 0 0 0 0 0 0 9 x 10 4

(537)

119


1 0 0 0 0 1 0 0 0 0 5 x 10 4 0

LO 0 0 5 x 10 4

(538)





(539)







these conditions



xf =

$d =

Aku

0

0 Akv

Cku o

Cu 0

Bku

0 B Xf

Xf = Ckxf

0

kv

Hf = Akxf - Bkyf

(540)

Vu Vv 0 Cv

Xf = CfXf



120


X

(541)

Ckxf















vibration case




tems

121






122





t o

1 2 Time (Orbits)



(b)

Figure 5-8



123

























124


























125











angles

















126

flexible tether






















based experiments


dynamics

127

BIBLIOGRAPHY
















128















129














130

6 1995 pp 373-384




















Letter V o l 2 No 1 1982 pp 48-52


131








132























133





w = Av (12)


4-1 ltL3gt j







equation Similarly

E = aAB + bCD (15)








notation

Wi = AjiVj (17)

134




tensor notation as

di-dx-j-1 ( L 8 )


f - a )


is given by

C = Ax)Bx) + A(x)Bx) (110)






13 Forcing Function








135


Mij = qsRv

ni^MtnrnRv

mj + Rv

niMtnTn^Rv

mj + Rv

niMtnmRv

mjs) (113)


-JPdMu 1 bdquo T x






dq dq dqt = dq^QP^




Fk(q^q^)=(RP

sk + R P n J q n ) ^ - - Q d k





(118)

136

















attitude angles

ci = on + 6

Pi = Pi (HI)

7t = 0



137



-S 0 c Pi

(II2)



(215)



(II3) Vi = I Pi i


where

and



(II4)

(II5)

(II6)

D1 = D D l D S v

52 = 31 + P(d2)m +T[d2

D3 = D2 + l2 + dX2Pii)m + T2i6x






138


KH = miDD + 2DiP






TV



d t dqred d(ired

with





139








th



respectively





in the ith link






F-Q mdash








ix




- $ T



qv---QtNT


adjacent links)



i + $DJi




tions respectively





LIST OF FIGURES


satellite system 2

























xi













without offset










without offset






x i i






















the local vertical








x i i i








xiv



















xv

1 INTRODUCTION























1


2

Satellite N


2


3



























4





tether





















5


lt mdash

lt Satellite


6


























7















ML+F = Q





be considered






8




























9



























10



























11

two methods[39]
























12





motion



modes










( L Q G L T - R )








13


21 Kinematics















frame



Di + Ti(i + f(i)) (21)

14

15


Ui nvi

wi I I i = 1

nzi bull

(22)



taken as 2 j - l

Hi(xiii)=[j) (23)




k (24)





ff(i) = $i(xili)5i(t) (25)


0 0

^i(xiJi) =


0

0 0

0

G 5R 3 x n f

(26)



16






( 3 d

dx~ + dT)^Xili^ ( 2 8 )

one arrives at











1 P i tch F[ = Cf ( a j )F 0



17

where

c f (A) =

-Sai Cai

0 0

and

C(7i ) =

0 -s0i] = 0 1 0

0 cpi

i 0 0 = 0

0

(211)

(212)

(213)








(214) SliCPi CliCPi









18



















(219)

that

(220)

19


20


cos(0i) 0 0 s i n ( 0 i ) 0 0

0 0 1

Dri

h (221)

= DTlDSv




1 1 1 1 J (222)

= D D l D 3 v


(223)



0 0 1

DTl = D D i = I d 3 i = 2N (224)


2N (225)








21





(226)





and

Ai = i(ti) - ^(t 0 -) = - ^ ( t 2 i ) (227)


V = lUli l^lL (228)

1 tf-tQ v y

A t = (229)


221 Kinetic energy




4 ^ ^ PM) T i $ 4 TiSi]i[t (231)

22

where

m

k J

x l (232)



rewritten as

1 T

(233)


Mt =



pound Sfttiqxnq

(234)




i=l (235)



(236) Mt =

0 0

0 Mt2

0

0 0

MH

0 0 0

0 0 0 bull MtN



23


Mti

m D D l


sym 0

0


0 Id nfj

0 m~

i = 1 3 5 N

(237)


gets

D D l

1 D D l = 1 0 0 0 D n

2 0 (238) 0 0 1










sym sym



tr(ffTai

TTaifi) tr(ffTai





(241)



24

and

j tr(A) =tr(^J A^j (243)



P(fi)Pii)dmi =

tr(Tai

TTaiImi) tTTai


TT7Jmi) sym sym sym

(244)


expanded is


sym sym


sym J z^dmi J


(245)






can be written as

V9i f dm f


dm

inn Lgti + T j ^ | (246)





(2-lt 7)

25






readily that



224 Strain energy












(249)




26




by





number of modes







EX^Ei+jEi (252)





experimentally [49]

27


jet = (254) wo-





(ii) Jo




dWd

^ = - i p = 2 4 ( 2 5 7 )

= 0 i = 13 N









28

system

Mq t)t+ F(q q t) = Q(q q t) (259)


















Let

e Unqxl (261) m


29











3 = 1



k=l



Qt Jfe=l

where

and


T fc-1 0 TkKAk

0 0 0 0 0 0 0 0 0 0 0 0 -

r T i - i 0 0 0 0 0 0 0 0 0 G

0 0 0 1

Qt = RpQ

G R n lt x l

(265)

(266)

G Unq x l (267)

(268)

30

where

RP

R 1 0 0 R R 2 0 R RP

2 R 3

0 0 0

R RP R PN _ 1 RN J

(269)




transformations



t h l ink as

D = A - l + Ti_iK + + K A i _ i ^ _ i


(271) (











31




t = Rvil (2-74)


Here

Rv


0 0 0

T3V nN-l

R bullN

e Mnqqxl

Ri =


e K n lt x l

RS

T i-1 0 0 o-0 0 0 0 0 0

0 0 0 1

G Mnq x l

(275)

(276)

(277)




(278)


0 0

0 0

0 0

G $lnq x l


(279)

pound = Rnjt + Rdq


32

where

and

- 0 0 0 RI 0 0

Rn = 0 0

0 0

Rf 0 0 0 rgtd K2 0

Rd = 0 0 Rl

0 0 0

RN-1 0

G 5R n w x l

0 0 0 e 5 R N ^ X L

(281)

(282)






where

D D T i


DTL 0 0 0 d 3 0 0 0 ID

0 0 nfj 0

RP =


2 R3

iJfDTTi i^

x l

0 0 0

R PN-I RN

(284)

(285)



33



0 0 0 0 0 0 0 0

(286)









M = RV MtRv


(289)

(290)

Inverting Eq(290)



34


Mr1

h 0 0

0 0

0 0

0 0 0

0

0

0

tN

(292)



operation








Mq + f = P C A (293)




pcTM-pc

= ( dc + PdegTM-^f) (294)




35


















nu ^5 Qk=E Pern ^ (2-97)

m=l deg Q k






36






Defining


= d + dfY + T 3 f c m 3



(298)

(299)



(2100)

as i - 2

k=l i1 cm (2101)



WjRi A R i (2102)




1 Q

Q3

3 -1

T 3 T t 2 (2103)

37

Satellite 3

mdash gt -raquo cm

D1=d1

mdash gt

a mdash gt

o


38




Q

~Qgt1

0

0

( T ^ 2 ) (2104)







(2105)

where





3

Qk = ^ F e i

i=l

d

dqk (2106)




39


40

be written as


QiMmi



(2107)

(2108)



Q

Q T 3 0 Q T 5 0

0 0 0

0 lt33T3 Qyen Q5

3T5 0

0 0 0 Q4T5 0

0 0 0 Q5

5T5

M M I

Ti 2 m 3

Quu (2109)


41










numerically














42















by letting

thus

X - ^ ^ f ^ ) =ltbullbull) (3 -2 )







43








plott ing package



















44

INIT

DGEAR PARAM

IC amp SYS PARAM

MAIN

lt

DGEAR

lt

FCN

OUTPUT



VIBRATION

r OFFS DYNA

ET MICS


45









longer stored




the system











46

h =

1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760



0 400 0 0 0 400












fcm3 = 10 0T










47




OOEOh

-50E-12

-10E-11 I i i



48



(a)

20E6

10E6r-

00E0

-10E6

-20E6


-

1 I I I 1 I I

I I

AP e ~_

i i 1 i i i i

00E0

UJ -50E-12 U J

lt


-10E-11h

(b) Time (Orbits)


49


























50

a p(0) = 2deg 0(0) = 2C

6(0) = 08 m

0(0) = 001 m


D p y = 0 D p 2 = 0

a oo

c -ooo -0 01


51






^MAAAAAAAWVWWWWVVVW

V bull i bull i I i J 00 05 10 15 20


Time (Orbits)


52
























53


1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760

h =

h = 200 0 0

0 400 0 0 0 400















mdash







54



55




























56







57







(b) Time (Orbits)


58










by the coupling

















59








60



Time (Orbits)



Time (Orbits)



0 1 2 3 4 5 (b) Time (Orbits)


61





Tether Pitch Angle


Tether Roll Angle

CD

edT

1 2 3 Time (Orbits)




CD

C O

(a)

Figure 4-




62



0 1 2 3 4 5







0 1 2 3 4 5 (b) Time (Orbits)


63



2 3 4 Time (Orbits)

Tether Pitch Angle


Tether Roll Angle



2 3 4 Time (Orbits)


(a)

Figure 4-


2 3 4 Time (Orbits)


64




(b) Time (Orbits)


65










V0 = 146 ms
















66



i i i i i i i i i i i 0 1 2 3 4 5 0 1 2 3 4 5






67



0 1 2 3 4 5


(b) Time (Orbits)


68




E





Tether Pitch Angle


Tether Roll Angle

CD

oo ax


_ 1 _ L





-50

(a) 1 2 3 4


Time (Orbits)


69




to

-02

(b)

Figure 4-7

2 3 Time (Orbits)


0 1 1 bull 1 1



5

1 1 1 1 1 i i | i i | ^

_ i i i i L -I



70












roll

45 Tether Retrieval










and subsatellite



71




E


1 2 3 Time (Orbits)




Tether Pitch Angle

1 2 3 Time (Orbits)

Tether Roll Angle

i i i i J I I i _





D )

T J ^ 0 E 0 eg

CQ

-3E-3

(a) 1 2 3 4


Time (Orbits)


7 2





n 1 1 1 f

I I I _j I I I I I 1 1 1 1 1 2 3 4 5



Time (Orbits)


(b)

Figure 4-8

73




(a) Time (Orbits)



i i i i i i i i i i I 0 1 2 3






04- -



74




E 10 to


1 2 Time (Orbits)


(b) Time (Orbits)


75

























76


77



o 1 2 Time (Orbits)


0 1 2 Time (Orbits)






78


p1(0)=p2(0)=(33(0)=pt1(0)=pt2(0)=10





79



























80






i i i i i 0 1 2 3 4 5










81






82



























83



I- I i t i i i i I H 0 1 2 0 1 2





84



laquo = 5 E - 3 |

000 002



Time (Orbits)


85




E

2000

cu T3

2000

co CD

2000

C D CD

T J


1 2 Time (Orbits)


cn CD

33 cdl




1 2 Time (Orbits)




86



5E-3h

co

0E0

025

000 E to

-025

002

-002

(b)

Figure 4-14




Time (Orbits)


87






















88





89


1E-2

to

OEO

E 00

to


005



Time (Orbits)


90





91




92

























93


present case













mance











94





by Eq(2109)

Lett ing

~fr = fr~ QuU (52)


ir = s(frMrdc) (53)



mdash = Fr - Qru





Ire = Frc ~~ Qrcu

(56)

mdash vrci






an(^ qrcd

95









-In ) 0 5 V T ^ r (59)


V (510)








qrcd 37Vxl (511)

96


e U 3 N x l (512)

E K 3 7 V x l (513)

















97

and

o 0

Qrcj mdash

o o

0 0

Oiit) 0 0

1

N

N

1deg I o







20

CO CD

T J

cdT

cT

- mdash mdash 1 1 1 1 1 I - 1

Pitch -

A Roll -

A -

1 bull bull

-

1 Time (Orbits)


1 2 Time (Orbits)


gt to N

to


145h

(a) Time (Orbits)


98

1





99
























reduced


100



c n

T J

cdT




c n CD

cdT a

o T J

CQ

- - I 1 1 1 I 1 1 1 1

P i t c h

R o l l

bull

1 -

0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


E gt

10 N

CO


145h

(a) Time (Orbits)


101






102






CD

eg CQ

0 1 2 Time (Orbits)


1 2 Time (Orbits)


gt CO


145h

(a) Time (Orbits)


103





0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


-07 o 1


1E-5

0E0F

Time (Orbits)


104





o 1 2 Time (Orbits)


0 1 2 Time (Orbits)


CD CD

CM CO


to

150

100

1 2 3 0 1 2


to

(a)

50

2 3 Time (Orbits)


105





2 0 h

E 2L


0 5

1 2 3



1 2


1 2 3 Time (Orbits)


2 E - 5

1 2 3


1 2


1 E - 5

E

O E O

Time (Orbits)


106

























107





0 1 2 Time (Orbits)


CO CD

cdT

CO CD

T J CM

CO

1 2 Time (Orbits)



150

100



CO

(a)

x 50

2 3 Time (Orbits)


108




E

1 2 Time (Orbits)


2E-5

1 2 Time (Orbits)


OEOft

1 2 3 Time (Orbits)


1 2 3 Time (Orbits)


1 2 Time (Orbits)


1E-5 E z

2

0E0



109















are carried out











110












efficient












111






91 remains constant








by Ms

1 gives

qr = -Ms 1Csqr - Ms

lKsqr



Pu t t ing

v =

+

-M~lCs -M~Cd -1

0

-Mg~lMd

Id

0 v + -M~lKs -M~lKd

0 0

-M-lfs

0



(518)

(519)

x =Ax + Bud + Fd (520)

112

where

A = MC MK Id12 o

24x24

and

Let

B = ^ U 5 R 2 4 X 3 -

Fd = I FJ ) e f t 2 4 1

mdash _ i _ -5

(521)

(522)

(523)

(524)







S=Ax + Bud (526)






K(o)sx

V

dx ~ dXQ dy ~ dyQ

d z - d z0 J

Jy

dx da XQ vo

d z - dZQ ) dy dyQ

(527)

113


























114

u

MODEL UNCERTAINTY

SYSTEM PLANT

y

u LQR CONTROLLER

X KBF ESTIMATOR

u

y

LQG COMPENSATOR

PLANT

X = f v ( X t )

COMPENSATOR

x = A k x - B k y

ud

= C k f

y


115














summarized below



(528)


0 0 $u(lt) 0 0 0 0 1

0 0 1 0 0 0 0 1

(529)


components of xu


Qu =

bull1 0 0 o -0 1 0 0 0 0 1 0

0 0 0 10

(530)

116


selected as 1 0 0 0 0 1 0 0 0 0 4 0

LO 0 0 1


(531)

i o 0 15

(532)





where r is a scaler













117

100

co

gt CO

-50

-100 10


Target r = 5x10 5

r = 0 (LQG)


(a)


101 10


100

50

X3

gt 0

CO

-50

bull100 10




(b)


101 10


118


parts[63]



Dv = Cvxv


(534)

Cy

0 0 0 0 (0) 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0

h 0 0 0

0 0 0 lt ( 0 ) o 0

0 1 0 0 0 1

0 0 0

2 r

h 0 0

0 0 1 0 0 1

(535)


O f Xy


Qv

10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

(536)

mdash


1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 x 10 4 0

0 0 0 0 0 0 0 9 x 10 4

(537)

119


1 0 0 0 0 1 0 0 0 0 5 x 10 4 0

LO 0 0 5 x 10 4

(538)





(539)







these conditions



xf =

$d =

Aku

0

0 Akv

Cku o

Cu 0

Bku

0 B Xf

Xf = Ckxf

0

kv

Hf = Akxf - Bkyf

(540)

Vu Vv 0 Cv

Xf = CfXf



120


X

(541)

Ckxf















vibration case




tems

121






122





t o

1 2 Time (Orbits)



(b)

Figure 5-8



123

























124


























125











angles

















126

flexible tether






















based experiments


dynamics

127

BIBLIOGRAPHY
















128















129














130

6 1995 pp 373-384




















Letter V o l 2 No 1 1982 pp 48-52


131








132























133





w = Av (12)


4-1 ltL3gt j







equation Similarly

E = aAB + bCD (15)








notation

Wi = AjiVj (17)

134




tensor notation as

di-dx-j-1 ( L 8 )


f - a )


is given by

C = Ax)Bx) + A(x)Bx) (110)






13 Forcing Function








135


Mij = qsRv

ni^MtnrnRv

mj + Rv

niMtnTn^Rv

mj + Rv

niMtnmRv

mjs) (113)


-JPdMu 1 bdquo T x






dq dq dqt = dq^QP^




Fk(q^q^)=(RP

sk + R P n J q n ) ^ - - Q d k





(118)

136

















attitude angles

ci = on + 6

Pi = Pi (HI)

7t = 0



137



-S 0 c Pi

(II2)



(215)



(II3) Vi = I Pi i


where

and



(II4)

(II5)

(II6)

D1 = D D l D S v

52 = 31 + P(d2)m +T[d2

D3 = D2 + l2 + dX2Pii)m + T2i6x






138


KH = miDD + 2DiP






TV



d t dqred d(ired

with





139




- $ T



qv---QtNT


adjacent links)



i + $DJi




tions respectively





LIST OF FIGURES


satellite system 2

























xi













without offset










without offset






x i i






















the local vertical








x i i i








xiv



















xv

1 INTRODUCTION























1


2

Satellite N


2


3



























4





tether





















5


lt mdash

lt Satellite


6


























7















ML+F = Q





be considered






8




























9



























10



























11

two methods[39]
























12





motion



modes










( L Q G L T - R )








13


21 Kinematics















frame



Di + Ti(i + f(i)) (21)

14

15


Ui nvi

wi I I i = 1

nzi bull

(22)



taken as 2 j - l

Hi(xiii)=[j) (23)




k (24)





ff(i) = $i(xili)5i(t) (25)


0 0

^i(xiJi) =


0

0 0

0

G 5R 3 x n f

(26)



16






( 3 d

dx~ + dT)^Xili^ ( 2 8 )

one arrives at











1 P i tch F[ = Cf ( a j )F 0



17

where

c f (A) =

-Sai Cai

0 0

and

C(7i ) =

0 -s0i] = 0 1 0

0 cpi

i 0 0 = 0

0

(211)

(212)

(213)








(214) SliCPi CliCPi









18



















(219)

that

(220)

19


20


cos(0i) 0 0 s i n ( 0 i ) 0 0

0 0 1

Dri

h (221)

= DTlDSv




1 1 1 1 J (222)

= D D l D 3 v


(223)



0 0 1

DTl = D D i = I d 3 i = 2N (224)


2N (225)








21





(226)





and

Ai = i(ti) - ^(t 0 -) = - ^ ( t 2 i ) (227)


V = lUli l^lL (228)

1 tf-tQ v y

A t = (229)


221 Kinetic energy




4 ^ ^ PM) T i $ 4 TiSi]i[t (231)

22

where

m

k J

x l (232)



rewritten as

1 T

(233)


Mt =



pound Sfttiqxnq

(234)




i=l (235)



(236) Mt =

0 0

0 Mt2

0

0 0

MH

0 0 0

0 0 0 bull MtN



23


Mti

m D D l


sym 0

0


0 Id nfj

0 m~

i = 1 3 5 N

(237)


gets

D D l

1 D D l = 1 0 0 0 D n

2 0 (238) 0 0 1










sym sym



tr(ffTai

TTaifi) tr(ffTai





(241)



24

and

j tr(A) =tr(^J A^j (243)



P(fi)Pii)dmi =

tr(Tai

TTaiImi) tTTai


TT7Jmi) sym sym sym

(244)


expanded is


sym sym


sym J z^dmi J


(245)






can be written as

V9i f dm f


dm

inn Lgti + T j ^ | (246)





(2-lt 7)

25






readily that



224 Strain energy












(249)




26




by





number of modes







EX^Ei+jEi (252)





experimentally [49]

27


jet = (254) wo-





(ii) Jo




dWd

^ = - i p = 2 4 ( 2 5 7 )

= 0 i = 13 N









28

system

Mq t)t+ F(q q t) = Q(q q t) (259)


















Let

e Unqxl (261) m


29











3 = 1



k=l



Qt Jfe=l

where

and


T fc-1 0 TkKAk

0 0 0 0 0 0 0 0 0 0 0 0 -

r T i - i 0 0 0 0 0 0 0 0 0 G

0 0 0 1

Qt = RpQ

G R n lt x l

(265)

(266)

G Unq x l (267)

(268)

30

where

RP

R 1 0 0 R R 2 0 R RP

2 R 3

0 0 0

R RP R PN _ 1 RN J

(269)




transformations



t h l ink as

D = A - l + Ti_iK + + K A i _ i ^ _ i


(271) (











31




t = Rvil (2-74)


Here

Rv


0 0 0

T3V nN-l

R bullN

e Mnqqxl

Ri =


e K n lt x l

RS

T i-1 0 0 o-0 0 0 0 0 0

0 0 0 1

G Mnq x l

(275)

(276)

(277)




(278)


0 0

0 0

0 0

G $lnq x l


(279)

pound = Rnjt + Rdq


32

where

and

- 0 0 0 RI 0 0

Rn = 0 0

0 0

Rf 0 0 0 rgtd K2 0

Rd = 0 0 Rl

0 0 0

RN-1 0

G 5R n w x l

0 0 0 e 5 R N ^ X L

(281)

(282)






where

D D T i


DTL 0 0 0 d 3 0 0 0 ID

0 0 nfj 0

RP =


2 R3

iJfDTTi i^

x l

0 0 0

R PN-I RN

(284)

(285)



33



0 0 0 0 0 0 0 0

(286)









M = RV MtRv


(289)

(290)

Inverting Eq(290)



34


Mr1

h 0 0

0 0

0 0

0 0 0

0

0

0

tN

(292)



operation








Mq + f = P C A (293)




pcTM-pc

= ( dc + PdegTM-^f) (294)




35


















nu ^5 Qk=E Pern ^ (2-97)

m=l deg Q k






36






Defining


= d + dfY + T 3 f c m 3



(298)

(299)



(2100)

as i - 2

k=l i1 cm (2101)



WjRi A R i (2102)




1 Q

Q3

3 -1

T 3 T t 2 (2103)

37

Satellite 3

mdash gt -raquo cm

D1=d1

mdash gt

a mdash gt

o


38




Q

~Qgt1

0

0

( T ^ 2 ) (2104)







(2105)

where





3

Qk = ^ F e i

i=l

d

dqk (2106)




39


40

be written as


QiMmi



(2107)

(2108)



Q

Q T 3 0 Q T 5 0

0 0 0

0 lt33T3 Qyen Q5

3T5 0

0 0 0 Q4T5 0

0 0 0 Q5

5T5

M M I

Ti 2 m 3

Quu (2109)


41










numerically














42















by letting

thus

X - ^ ^ f ^ ) =ltbullbull) (3 -2 )







43








plott ing package



















44

INIT

DGEAR PARAM

IC amp SYS PARAM

MAIN

lt

DGEAR

lt

FCN

OUTPUT



VIBRATION

r OFFS DYNA

ET MICS


45









longer stored




the system











46

h =

1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760



0 400 0 0 0 400












fcm3 = 10 0T










47




OOEOh

-50E-12

-10E-11 I i i



48



(a)

20E6

10E6r-

00E0

-10E6

-20E6


-

1 I I I 1 I I

I I

AP e ~_

i i 1 i i i i

00E0

UJ -50E-12 U J

lt


-10E-11h

(b) Time (Orbits)


49


























50

a p(0) = 2deg 0(0) = 2C

6(0) = 08 m

0(0) = 001 m


D p y = 0 D p 2 = 0

a oo

c -ooo -0 01


51






^MAAAAAAAWVWWWWVVVW

V bull i bull i I i J 00 05 10 15 20


Time (Orbits)


52
























53


1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760

h =

h = 200 0 0

0 400 0 0 0 400















mdash







54



55




























56







57







(b) Time (Orbits)


58










by the coupling

















59








60



Time (Orbits)



Time (Orbits)



0 1 2 3 4 5 (b) Time (Orbits)


61





Tether Pitch Angle


Tether Roll Angle

CD

edT

1 2 3 Time (Orbits)




CD

C O

(a)

Figure 4-




62



0 1 2 3 4 5







0 1 2 3 4 5 (b) Time (Orbits)


63



2 3 4 Time (Orbits)

Tether Pitch Angle


Tether Roll Angle



2 3 4 Time (Orbits)


(a)

Figure 4-


2 3 4 Time (Orbits)


64




(b) Time (Orbits)


65










V0 = 146 ms
















66



i i i i i i i i i i i 0 1 2 3 4 5 0 1 2 3 4 5






67



0 1 2 3 4 5


(b) Time (Orbits)


68




E





Tether Pitch Angle


Tether Roll Angle

CD

oo ax


_ 1 _ L





-50

(a) 1 2 3 4


Time (Orbits)


69




to

-02

(b)

Figure 4-7

2 3 Time (Orbits)


0 1 1 bull 1 1



5

1 1 1 1 1 i i | i i | ^

_ i i i i L -I



70












roll

45 Tether Retrieval










and subsatellite



71




E


1 2 3 Time (Orbits)




Tether Pitch Angle

1 2 3 Time (Orbits)

Tether Roll Angle

i i i i J I I i _





D )

T J ^ 0 E 0 eg

CQ

-3E-3

(a) 1 2 3 4


Time (Orbits)


7 2





n 1 1 1 f

I I I _j I I I I I 1 1 1 1 1 2 3 4 5



Time (Orbits)


(b)

Figure 4-8

73




(a) Time (Orbits)



i i i i i i i i i i I 0 1 2 3






04- -



74




E 10 to


1 2 Time (Orbits)


(b) Time (Orbits)


75

























76


77



o 1 2 Time (Orbits)


0 1 2 Time (Orbits)






78


p1(0)=p2(0)=(33(0)=pt1(0)=pt2(0)=10





79



























80






i i i i i 0 1 2 3 4 5










81






82



























83



I- I i t i i i i I H 0 1 2 0 1 2





84



laquo = 5 E - 3 |

000 002



Time (Orbits)


85




E

2000

cu T3

2000

co CD

2000

C D CD

T J


1 2 Time (Orbits)


cn CD

33 cdl




1 2 Time (Orbits)




86



5E-3h

co

0E0

025

000 E to

-025

002

-002

(b)

Figure 4-14




Time (Orbits)


87






















88





89


1E-2

to

OEO

E 00

to


005



Time (Orbits)


90





91




92

























93


present case













mance











94





by Eq(2109)

Lett ing

~fr = fr~ QuU (52)


ir = s(frMrdc) (53)



mdash = Fr - Qru





Ire = Frc ~~ Qrcu

(56)

mdash vrci






an(^ qrcd

95









-In ) 0 5 V T ^ r (59)


V (510)








qrcd 37Vxl (511)

96


e U 3 N x l (512)

E K 3 7 V x l (513)

















97

and

o 0

Qrcj mdash

o o

0 0

Oiit) 0 0

1

N

N

1deg I o







20

CO CD

T J

cdT

cT

- mdash mdash 1 1 1 1 1 I - 1

Pitch -

A Roll -

A -

1 bull bull

-

1 Time (Orbits)


1 2 Time (Orbits)


gt to N

to


145h

(a) Time (Orbits)


98

1





99
























reduced


100



c n

T J

cdT




c n CD

cdT a

o T J

CQ

- - I 1 1 1 I 1 1 1 1

P i t c h

R o l l

bull

1 -

0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


E gt

10 N

CO


145h

(a) Time (Orbits)


101






102






CD

eg CQ

0 1 2 Time (Orbits)


1 2 Time (Orbits)


gt CO


145h

(a) Time (Orbits)


103





0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


0 1 2 Time (Orbits)


-07 o 1


1E-5

0E0F

Time (Orbits)


104





o 1 2 Time (Orbits)


0 1 2 Time (Orbits)


CD CD

CM CO


to

150

100

1 2 3 0 1 2


to

(a)

50

2 3 Time (Orbits)


105





2 0 h

E 2L


0 5

1 2 3



1 2


1 2 3 Time (Orbits)


2 E - 5

1 2 3


1 2


1 E - 5

E

O E O

Time (Orbits)


106

























107





0 1 2 Time (Orbits)


CO CD

cdT

CO CD

T J CM

CO

1 2 Time (Orbits)



150

100



CO

(a)

x 50

2 3 Time (Orbits)


108




E

1 2 Time (Orbits)


2E-5

1 2 Time (Orbits)


OEOft

1 2 3 Time (Orbits)


1 2 3 Time (Orbits)


1 2 Time (Orbits)


1E-5 E z

2

0E0



109















are carried out











110












efficient












111






91 remains constant








by Ms

1 gives

qr = -Ms 1Csqr - Ms

lKsqr



Pu t t ing

v =

+

-M~lCs -M~Cd -1

0

-Mg~lMd

Id

0 v + -M~lKs -M~lKd

0 0

-M-lfs

0



(518)

(519)

x =Ax + Bud + Fd (520)

112

where

A = MC MK Id12 o

24x24

and

Let

B = ^ U 5 R 2 4 X 3 -

Fd = I FJ ) e f t 2 4 1

mdash _ i _ -5

(521)

(522)

(523)

(524)







S=Ax + Bud (526)






K(o)sx

V

dx ~ dXQ dy ~ dyQ

d z - d z0 J

Jy

dx da XQ vo

d z - dZQ ) dy dyQ

(527)

113


























114

u

MODEL UNCERTAINTY

SYSTEM PLANT

y

u LQR CONTROLLER

X KBF ESTIMATOR

u

y

LQG COMPENSATOR

PLANT

X = f v ( X t )

COMPENSATOR

x = A k x - B k y

ud

= C k f

y


115














summarized below



(528)


0 0 $u(lt) 0 0 0 0 1

0 0 1 0 0 0 0 1

(529)


components of xu


Qu =

bull1 0 0 o -0 1 0 0 0 0 1 0

0 0 0 10

(530)

116


selected as 1 0 0 0 0 1 0 0 0 0 4 0

LO 0 0 1


(531)

i o 0 15

(532)





where r is a scaler













117

100

co

gt CO

-50

-100 10


Target r = 5x10 5

r = 0 (LQG)


(a)


101 10


100

50

X3

gt 0

CO

-50

bull100 10




(b)


101 10


118


parts[63]



Dv = Cvxv


(534)

Cy

0 0 0 0 (0) 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0

h 0 0 0

0 0 0 lt ( 0 ) o 0

0 1 0 0 0 1

0 0 0

2 r

h 0 0

0 0 1 0 0 1

(535)


O f Xy


Qv

10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

(536)

mdash


1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 x 10 4 0

0 0 0 0 0 0 0 9 x 10 4

(537)

119


1 0 0 0 0 1 0 0 0 0 5 x 10 4 0

LO 0 0 5 x 10 4

(538)





(539)







these conditions



xf =

$d =

Aku

0

0 Akv

Cku o

Cu 0

Bku

0 B Xf

Xf = Ckxf

0

kv

Hf = Akxf - Bkyf

(540)

Vu Vv 0 Cv

Xf = CfXf



120


X

(541)

Ckxf















vibration case




tems

121






122





t o

1 2 Time (Orbits)



(b)

Figure 5-8



123

























124


























125











angles

















126

flexible tether






















based experiments


dynamics

127

BIBLIOGRAPHY
















128















129














130

6 1995 pp 373-384




















Letter V o l 2 No 1 1982 pp 48-52


131








132























133





w = Av (12)


4-1 ltL3gt j







equation Similarly

E = aAB + bCD (15)








notation

Wi = AjiVj (17)

134




tensor notation as

di-dx-j-1 ( L 8 )


f - a )


is given by

C = Ax)Bx) + A(x)Bx) (110)






13 Forcing Function








135


Mij = qsRv

ni^MtnrnRv

mj + Rv

niMtnTn^Rv

mj + Rv

niMtnmRv

mjs) (113)


-JPdMu 1 bdquo T x






dq dq dqt = dq^QP^




Fk(q^q^)=(RP

sk + R P n J q n ) ^ - - Q d k





(118)

136

















attitude angles

ci = on + 6

Pi = Pi (HI)

7t = 0



137



-S 0 c Pi

(II2)



(215)



(II3) Vi = I Pi i


where

and



(II4)

(II5)

(II6)

D1 = D D l D S v

52 = 31 + P(d2)m +T[d2

D3 = D2 + l2 + dX2Pii)m + T2i6x






138


KH = miDD + 2DiP






TV



d t dqred d(ired

with





139

DYNAMICS AND CONTROL OF MULTIBODY TETHERED SYSTEMS …

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