Ph.D. Thesis: 
The Dynamics of Tethers and Space-webs 


David J. McKenzie, BEng 


Submitted in fulfilment of the requirements for the Degree of Doctor of Philosophy 


Departments of Mechanical and Aeronautical Engineering, 
Faculty of Engineering, 
University of Glasgow, G12 8QQ, Scotland, UK 


January 2010 


. 
David McKenzie, 2003-2010 

Contents 


Abstract 15 

Acknowledgements 16 

1 Introduction 19 

1.1 Overviewof thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 

2 Literature review 23 

2.1 Earlypioneers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 

2.2 Tetherfundamentals ........................... 25 


2.2.1 Fundamental tetherconcepts .................. 26 


2.2.2 Non-rotating tethers ....................... 26 


2.2.3 Momentumexchange ....................... 27 


2.2.4 Motorized momentum exchange tethers . . . . . . . . . . . . . 29 

2.2.5 Staging with momentumexchange . . . . . . . . . . . . . . . 30 

2.2.6 Deploying and recovery of tethers . . . . . . . . . . . . . . . . 31 

2.2.7 Captureand rendezvousof tethers . . . . . . . . . . . . . . . 33 

2.2.8 WSB trajectories ......................... 33 


2.3 Tetherderived structures. . . . . . . . . . . . . . . . . . . . . . . . . 35 

2 



2.3.1 Deployed spacestructures .................... 35 


2.3.2 Space-webs ............................ 36 


3 Tether modelling 38 

3.1 Constructing the equations of motion of a system . . . . . . . . . . . 38 

3.1.1 Validating codemodules ..................... 39 


3.2 Centreof massmodelling . . . . . . . . . . . . . . . . . . . . . . . . . 40 

3.2.1 Masspoints&body selection .................. 40 


3.2.2 Centreof masscalculations ................... 41 


3.2.3 Positionsof masspoints ..................... 43 


3.3 Constructing Lagrange’sequations . . . . . . . . . . . . . . . . . . . 48 

3.3.1 Energy modelling ......................... 49 


3.3.2 Choosing generalised coordinate systems . . . . . . . . . . . . 50 

3.4 Non-conservativeforces. . . . . . . . . . . . . . . . . . . . . . . . . . 51 

3.5 Rotationsystems ............................. 52 


3.5.1 Rotationsummary ........................ 53 


3.5.2 YZRotations ........................... 55 


3.5.3 ZYRotations ........................... 56 


3.5.4 Comparing the equations – position equations . . . . . . . . . 59 

3.5.5 Comparing the equations – kinetic energies . . . . . . . . . . . 59 

3.5.6 Comparing the equations – potential energies . . . . . . . . . 60 

3.5.7 YZ rotationininertial axes ................... 61 


3.5.8 Singularities in the rotational systems . . . . . . . . . . . . . . 61 

3.5.9 Practical uses of the different equations . . . . . . . . . . . . . 62 

3 


3.6 Numericalintegrationtechniques .................... 62 


4 Dynamics of tethers with inclination 
64 

4.1 Additionofinclinationtotethermodel . . . . . . . . . . . . . . . . . 65 

4.1.1 5degreeoffreedommodel .................... 66 


4.2 Constructing theequationsof motion . . . . . . . . . . . . . . . . . . 67 

4.2.1 Rotations in the local coordinate system . . . . . . . . . . . . 68 

4.2.2 Lagrange’sequations ....................... 71 


4.2.3 Lagrange’sequations – local system . . . . . . . . . . . . . . . 71 

4.2.4 Lagrange’s equations – global system . . . . . . . . . . . . . . 75 

4.2.5 Non-conservativeforces. . . . . . . . . . . . . . . . . . . . . . 77 

4.3 Analysisof tetheronaninclined orbit . . . . . . . . . . . . . . . . . 80 

4.3.0.1 Behaviour of R, . 
and . 
with time .......... 81 


4.3.0.2 Behaviour of a 
with time ............... 81 


4.3.0.3 Behaviour of .a 
with time ............... 82 


4.3.0.4 
Behaviour of a 
with . 
. . . . . . . . . . . . . . . . . 82 
.¨ 

4.3.0.5 Behaviour of  , 
 , 
. 
. . . . . . . . . . . . . . . . . 83 

4.3.0.6 Behaviour of , 
, 
.¨. . . . . . . . . . . . . . . . . . 83 

4.3.0.7 Behaviour of 
V 
with time .............. 84 


4.3.0.8 Behaviour of tension with time . . . . . . . . . . . . 84 

4.3.0.9 Behaviourof stresswith time . . . . . . . . . . . . . 84 

4.3.1 Analysisof tensionand stressintether . . . . . . . . . . . . . 85 

4.4 Tetherreleaseaiming 
. . . . . . . . . . . . . . . . . . . . . . . . . . . 86 

4.4.1 Tethererroranalysis ....................... 88 


4 


4.4.2 Lunarcapture. . . . . . . . . . . . . . . . . . . . . . . . . . . 90 

4.5 A demonstration Lunar mission with a tether launch . . . . . . . . . 91 

4.5.1 Spacecraftsizing ......................... 91 


4.5.2 Analysis .............................. 93 


4.6 Conclusions ................................ 94 


4.7 Inclinationplots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 

5 Dynamics of tethers with length deployment 108 

5.1 Deploymentand recovery withtheMMET . . . . . . . . . . . . . . .108 

5.2 Addition of length as a generalised coordinate . . . . . . . . . . . . . 109 

5.3 Constructing theequationsof motion . . . . . . . . . . . . . . . . . .110 

5.4 Non-conservativeforces. . . . . . . . . . . . . . . . . . . . . . . . . .115 

5.4.1 Tetherbraking ..........................116 


5.5 Analysisoflengthdeploymentof tether . . . . . . . . . . . . . . . . .117 

5.5.1 Discussion of results for deployment . . . . . . . . . . . . . . . 118 

5.6 Refining theequationsof motion ....................120 


5.7 Recovery of tethers ............................123 


5.7.1 Recoveryprofile . . . . . . . . . . . . . . . . . . . . . . . . . .124 

5.7.2 Theequationsof motionforrecovery . . . . . . . . . . . . . .125 

5.7.3 Solving the equations of motion for recovery . . . . . . . . . . 126 

5.8 Deployment and recovery on an elliptical orbit . . . . . . . . . . . . . 128 

5.9 Centreof massmovement ........................130 


5.10Conclusions ................................131 


5.11Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132 

5 



5.11.1 Linearkineticenergy .......................143 


6 Dynamics of space-webs 144 

6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144 

6.2 Modelling methodology . . . . . . . . . . . . . . . . . . . . . . . . . .146 

6.3 Web meshing ...............................146 


6.3.1 Web structure. . . . . . . . . . . . . . . . . . . . . . . . . . .146 

6.3.2 Dividing theweb .........................147 


6.3.3 Webdivisions ...........................148 


6.4 Equationsof motion ...........................149 


6.4.1 Centreof massmodelling . . . . . . . . . . . . . . . . . . . . .149 

6.4.2 Rotations .............................150 


6.4.3 Positionequations ........................152 


6.5 Energy modelling .............................153 


6.5.1 Virtual workterms ........................154 


6.5.2 SimplifyingAssumptions .....................155 


6.5.3 RobotPositionModelling ....................156 


6.6 Space-webdynamics ...........................157 


6.6.1 Dynamical simulation. . . . . . . . . . . . . . . . . . . . . . .157 

6.7 Investigating the stability of the space-web . . . . . . . . . . . . . . . 158 

6.7.1 Massof theweb . . . . . . . . . . . . . . . . . . . . . . . . . .158 

6.7.2 Robotmass ............................159 


6.7.3 Web asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . .160 

6.7.4 Robotcrawl velocity .......................162 


6 


6.8 Statistical investigation into stability . . . . . . . . . . . . . . . . . . 162 

6.8.1 Massesof theweb and satellite . . . . . . . . . . . . . . . . .163 

6.8.2 Massesof theweb and robot. . . . . . . . . . . . . . . . . . .164 

6.8.3 Massof theweb and angularvelocity . . . . . . . . . . . . . .165 

6.8.4 Massof therobotand angularvelocity . . . . . . . . . . . . .167 

6.9 Conclusionsand recommendations. . . . . . . . . . . . . . . . . . . .168 

6.9.1 Conclusions ............................168 


6.9.2 Recommendations. . . . . . . . . . . . . . . . . . . . . . . . .168 

7 Conclusions 169 

7.1 Tetherrotations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169 

7.2 Tetherswithinclination .........................169 


7.3 Deploymentand recovery of tethers . . . . . . . . . . . . . . . . . . .170 

7.4 Space-webs ................................170 


7.5 Furtherstudy ...............................171 


8 Bibliography 172 

Glossary 183 

A Equations of motion with inclination 184 

B Space-webs – additional graphics 188 

B.1 Case 1 – CoM plots of stability while increasing web mass . . . . . . 188 

B.2 Case 1 – CoM plots of stability while increasing robot mass . . . . . . 190 

B.3 CoMplotsof stability whilechanging robotpaths . . . . . . . . . . .191 

7 



B.4 Case 1 – CoM plots of stability while decreasing robot velocity . . . . 192 
C Space-webs – additional data 195 
D Material strengths 200 
E Motor properties 202 

8 



List of Figures 


1.1 
Diagramof theMMET. ......................... 20 
2.1 
Tether length increment 
L, from [Cosmo and Lorenzini, 1997, p175] 27 
3.1 
Symmetricaldumbbelltether ...................... 41 
3.2 
Symmetricaldumbbelltetheronacircularorbit(solid). Afterpayload 
release,thefacility systempath atapogee(dash) andpayload 
atperigee(dots) ............................. 42 
3.3 
Asymmetricaldumbbellininertial axes . . . . . . . . . . . . . . . . . 44 
3.4 
Local coordinateconstruction ...................... 45 
3.5 
Local coordinate construction after rotation through angle . 
about 
thefacility mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 
3.6 
Local coordinate construction after translation from the facility to 
theCoM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 
3.7 
Local coordinate construction after translation from the CoM to the 
inertial coordinatesystem ........................ 46 
3.8 
Local coordinate construction after rotation through . 
about the inertial 
coordinatesystem ......................... 47 
9 



3.9 
Rotation sequence for YZ rotation: Rotating a 
about Y-axis then 
. 
about Z-axis. a 
is shown as a negative rotation here for ease of 
visualisation. ............................... 57 
3.10 Rotation sequence for ZY rotation.. Rotating . 
about Z-axis then 
a 
about Y-axis. a 
is shown as a negative rotation here for ease of 
visualisation. ............................... 58 
4.1 
Ephemeris of Lunar orbital inclination [JPL, 2008] . . . . . . . . . . . 65 
4.2 
Rotation sequence for YZ rotation with inclination. a 
is shown as a 
negative rotation here for ease of visualisation. . . . . . . . . . . . . . 69 
4.3 
RotationsequenceforYZrotationwithinclination. a 
and . 
are shown 
as negative rotations here for ease of visualisation. . . . . . . . . . . . 70 
4.4 
Eclipsecheckingfunction(nottoscale) . . . . . . . . . . . . . . . . . 79 
4.5 
Sample plot of the Eclipse function in Mathematica . . . . . . . . . . 79 
4.6 
Diagram showing an example of the WSB trajectory, from ESA Bulletin103:[
Biesbroek andJanin,2000]. . . . . . . . . . . . . . . . . . 87 
4.7 
Change in apogee of the payload after release from the tether system 
after one half of an orbit compared to small changes of system 
parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 
4.8 
Change in apogee and inclination of the payload after release from 
the tether system after onehalf of an orbit compared to small changes 
of systemparameters ........................... 96 
4.9 
The orbital debris environment from LEO to GEO, from [Wikipedia, 
2008] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 
4.10 Plots of R, . 
and . 
vs. timeforthefivecases. . . . . . . . . . . . . . 99 
4.11 Plots of a 
vs. timeforthefivecases. ..................100 
10 


4.12 Plots of .a 
vs. timeforthefivecases. ..................101 


4.13 Plots of a 
vs. . 
forthefivecases. ....................102 


4.14 Phase plots of . 
forthefivecases.. . . . . . . . . . . . . . . . . . . .103 

4.15 Phase plots of a 
forthefivecases.. . . . . . . . . . . . . . . . . . . .104 

4.16 Plots of 
V 
vs. timeforthefivecases. .................105 


4.17PlotsofTensionvs. timeforthefivecases. . . . . . . . . . . . . . . .106 

4.18PlotsofStressvs. timeforthefivecases. . . . . . . . . . . . . . . . .107 

5.1 Discretised tether diagram with n 
=5masspoints. . . . . . . . . . .112 

5.2 InitialMMETSpin-up ..........................132 


5.3 InitialMMETSpin-up ..........................133 


5.4 InitialMMETSpin-up ..........................133 


5.5 Initial MMET Spin-up, discretized tether mass model . . . . . . . . . 134 

5.6 Initial MMET Spin-up, discretized tether mass model . . . . . . . . . 135 

5.7 Initial MMET Spin-up, discretized tether mass model . . . . . . . . . 136 

5.8 MMET reelin,firstphase ........................136 


5.9 MMET reelin,firstphase ........................137 


5.10MMET reelin,firstphase ........................138 


5.11MMET reelin,secondphase. . . . . . . . . . . . . . . . . . . . . . .138 

5.12MMET reelin,secondphase. . . . . . . . . . . . . . . . . . . . . . .139 

5.13MMET reelin,secondphase. . . . . . . . . . . . . . . . . . . . . . .140 

5.14MMETdeploymentwith elliptical orbit . . . . . . . . . . . . . . . . .140 

5.15MMET recovery with elliptical orbit . . . . . . . . . . . . . . . . . .141 

5.16 MMET deployment, centre of mass movements . . . . . . . . . . . . . 141 

11 


5.17 MMET recovery, centre of mass movements . . . . . . . . . . . . . . 142 

5.18 MMET recovery, position and acceleration of centre of mass . . . . . 142 

6.1 Concept rendering of the space-web with spiderbots on the surface. . 145 

6.2 0divisions .................................148 


6.3 1division .................................148 


6.4 2divisions .................................148 


6.5 3sub-spans – 0divs ...........................148 


6.6 3sub-spans – 1div ............................148 


6.7 3sub-spans – 2divs ...........................148 


6.8 4sub-spans – 1div ............................148 


6.9 5sub-spans – 2divs ...........................148 


6.106 sub-spans – 3divs ...........................148 


6.11 Simplified space-web layout with s 
sub-spans. . . . . . . . . . . . . .149 

6.12Midpointlocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .150 

6.13 Space-web diagram with 3 sub-spans shown in inertial space. Note: 
thelocal axesareintheY-Zplane. . . . . . . . . . . . . . . . . . . .152 

6.14 Contours of CoM movement while varying web and satellite mass . . 164 

6.15 Contours of CoM movement while varying web and robot mass . . . . 165 

6.16 Contours of CoM movement while varying web mass and @. 
. . . . .166 

@t 


6.17 Contours of CoM movement while varying robot mass and @. 
. . . .167 

@t 


B.1 mW 
eb 
=27kg 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .188 

B.2 mW 
eb 
=33.75kg 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .188 

B.3 mW 
eb 
=37.8kg 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .189 

12 


B.4 mW 
eb 
=40.5kg 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .189 

B.5 mW 
eb 
=44.82kg 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .189 

B.6 mW 
eb 
=47.25kg 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .189 

B.7 mW 
eb 
=54kg 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .189 

B.8 mW 
eb 
=67.5kg 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .189 

B.9 Mrobot 
=1kg 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .190 

B.10 Mrobot 
=10kg 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .190 

B.11 Mrobot 
=15kg 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .190 

B.12 Mrobot 
=16kg 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .190 

B.13 Mrobot 
=20kg 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .190 

B.14 Mrobot 
=100kg 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .190 

B.15Case1 ...................................191 


B.16Case2 ...................................191 


B.17Case3 ...................................191 


B.18Case4 ...................................191 


B.19Case5 ...................................191 


B.20Case6 ...................................191 


B.21 Vrobot 
=10.0m/s 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .192 

B.22 Vrobot 
=2.0m/s 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192 

B.23 Vrobot 
=1.0m/s 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192 

B.24 Vrobot 
=0.2m/s 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192 

B.25 Vrobot 
=0.1m/s 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192 

B.26 Case 1 – 3 robots moving symmetrically round the web perimeter . . 193 

B.27 Case 6 – 1 robot moving asymmetrically along first sub-span . . . . . 194 
13 


List of Tables 

3.1 Orderof rotationsin3D and2D . . . . . . . . . . . . . . . . . . . . . 55 

4.1 Reference table with figure and corresponding rotation . . . . . . . . 68 

4.2 Initial conditions for the inclination numerical solutions. . . . . . . . 80 

4.3 Initial conditions for the inclination numerical solutions. . . . . . . . 98 

C.1 
Influenceof web massonmaximumCoM . . . . . . . . . . . . . . . .196 

C.2 
Influence of symmetrical robot movement on maximum CoM . . . . . 196 

C.3 
Influence of simulation time(effectively robot speed across web) on 
maximumCoM ..............................196 

C.4 
Influenceof robotmassonmaximumCoM . . . . . . . . . . . . . . .197 

C.5 
Statistical investigation into stability . . . . . . . . . . . . . . . . . . 198 

C.6 
Results of the statistical investigation into stability . . . . . . . . . . 199 

D.1 
Propertiesof selected materials . . . . . . . . . . . . . . . . . . . . .201 

E.1 
PropertiesofGEMotor5BC49JB1115 . . . . . . . . . . . . . . . . .202 

14 



Abstract 

The thesis ‘The dynamics of tethers and space-webs’ investigates the motion of the 
Motorized Momentum Exchange Tether (MMET) on an inclined orbit, and while 
deploying and retracting symmetricpayloads. TheMMET system isused asabasis 
for examiningthe stability of space-webs using atriangular structure oftethers while 
rotating. The motion of small robots is introduced as they move on the space-web, 
and their motions are found to influence the behaviour of the structure. Several 
methodsof limiting thedestabilisinginfluencesof therobotsareconsidered,and are 
found to stabilise the web in most circumstances. 

Astructured methodfordescribing therotationsof atethersystem isoutlined,and 
different rotational systems are compared. This lays the foundation for the further 
chapters examining MMET dynamics on an inclined orbit and while deploying and 
recovering the payloads. Lagrange’s equations are generated for the three cases, 
and are solved using standard numerical integration techniques. To emphasise the 
practical uses of the MMET system, several missions are analysed by using the 
system as a re-usable launcher for micro-satellite payloads. 

15 



Acknowledgements 

I would like to thank the EPSRC for providing a doctoral training grant, enabling 
the research for this thesis. 

I would like to thank the staff at the University of Glasgow for their thoughts and 
discussions. Inparticular,DrMaxVasileforhissharp insights,DrDavidForehand 
for discussions on many interesting problems and Dr Donald Ballance and Dr Ian 
Watson for an introduction into teaching. Needless to say, I am also thankful for 
thehard work of alltheadministrativestaff, inparticularMarilynDunlop andJane 
Livingston. 

I would like to thank Prof. Matthew Cartmell for the chance to study such an 
interesting problem, for his kindness and patience, for his supervision, and for enthusing 
me with thejoys of research. 

I would like to thank Colin McInnes for encouraging me to study space systems 
engineering. The love of space research, fostered under his guidance, played a big 
part in my decision to undertake a Ph.D.. 

I would like to thank Chris Draper for the long and interesting conversations we 
had while sharing an office, and for his enthusiasm in general. 

Iwould liketothank thestaffatESAACTfortheopportunity tostudy space-web 
dynamics. 

I would like to thank my friends at pausegaming.com, for the camaraderie. 

I would like to thank my families for their support. My dad and mum for encour


16 



aging my talents inmathematics, engineering andLEGOfromavery earlyageright 
through to the final chapter of this thesis. 

I would like to thank my very good friends: Raj, Aaron, Ailsa, Dougie, George, 
Rhys, and of course,Pob. Oursmallparty seemstohavegained afewmembers and 
lost a few, but we seem to have weathered the Storm: good friends are as valuable 
as gold. 

I would like to thank Joanna, for her kindness and encouragement, for her love 
and support, and for a thousand things too big and small to mention. 

17 



Declaration of Authorship 

The author hereby declares that this thesis is a result of the work carried out in 
the Department of Mechanical Engineering at the University of Glasgow during 
the period between October 2003 and January 2010. This thesis is an original 
contribution by the author unless otherwise indicated. 

David McKenzie 

January 2010 

18 



Chapter 1 
Introduction 

The field of space research is perhaps unique, in that many novel, different and 
interesting concepts are commonly developed from a seed of an idea to a satellite 
in a relatively short space of time. Additionally, mankind has been active in space 
foravery shorttime,buthasmadeastoundingprogresses inunderstanding inboth 
technical and social fields as a direct result of this. 

The cost of these endeavours has been equally astounding, with many nations 
contributing significant fractions of their GDP towards developing and maintaining 
a space presence. Equally, all nations with a space presence continually re-assess 
theircapability –politicianswantdevelopmentsfaster,accountantswanteverything 
donecheaper,scientistswanttodobetter–asadirect result,novelideascanflourish. 

TheMotorizedMomentumExchangeTether(MMET) isoneofthese ideas,with 
the potential to provide cheaper on-orbit launch costs and a lighter mass when 
compared to conventional rocket technologies. 

A diagram of the MMET is shown in Figure 1.1. A payload mass is connected 
to a large central facility mass with a high strength tether. A symmetrical mass 
is attached to the opposite side of the facility and these two tethers and payloads 

19 



formthepayloadarm. Afurthertwotethersarejoinedtothefacility to increase 
the inertia of facility, these are called the stator arm, and are similar in design to 
the payload arm. 

payload 2 
facility 
stator 1 
stator 2 
payload 1 
Figure 1.1: Diagram of the MMET. 

As with all alternative ideas, the MMET is best suited to a particular mission 
profile. With the MMET, its strongest advantage is that it is reusable, and may be 
employed to launch satellites and payloads as long as the tether is intact. 

Unfortunately, the MMET does have drawbacks, the four tethers are susceptible 
to severing from orbital debris, a symmetrical release strategy is required for each 
payload release and the risk of interference between the payload and stator arms is 
high. Additionally,themaximumorbital velocity thepayload canachieveis limited 
by the strength of the tethers. 

With these in mind, a large body of research into the performance of the MMET, 
momentum exchange tethers, dumbbell tethers, and tethers in general has revealed 

20 



that there are definite advantages in employing these as a tool to enhance or enable 
many satellite missions. As with all tools, space-or Earth-based alike, it is 
imperativeto usethe righttoolforthe rightjob. 

The research into momentum exchange tethers has been focussed on constructing 
mathematical models and analysing these models to determine the performance 
and stability, to avoid any potential weaknesses and to gain a better insight into 
the fundamental properties of the tether systems. Hardware testing is employed 
whereverpractical(andfunds allow!) to validatethese models. 

Several techniques are available to build these mathematical models. These include 
Newtonian based techniques, using forces and accelerations, and energy based 
techniques using Lagrangian formulations. 

This thesis will develop a mathematical model of the MMET using a Lagrangian 
based approach to further analyse the performance of the system in two key areas: 
on an inclined orbit, and while deploying and recovering the tether. 

A new structure – the space-web – is proposed and analysed, composed of a net 
overlaid on a backbone of a rotating tether system. This can be employed as a base 
on which to build large space structures such as solar sails and interferometers. 

1.1 Overview of thesis 
Chapter2 criticallyreviewsthecurrent literatureandgivesanoverviewof thewide 
range of tether systems. 

Chapter 3 introduces the rotational methods that underpin the rotational tether 
system and rotational systems are examined and compared. A method of constructing 
Lagrange’s equations for a rotating tether is outlined and the tether system is 
mathematically constructed from its individual components. 

Chapter 4 contains an analysis of the MMET dynamics on an inclined orbit, with 

21 



in-plane and out-of-plane motions. A demonstration of the tether’s capabilities is 
shown with an exemplar mission using aWeakStabilityBoundary(WSB)trajectory 
from Medium Earth Orbit(MEO) to the Moon. 

Chapter 5 studies the MMET while deploying and recovering the payload and 
tethers. A mathematical model of the MMET system undergoing deployment is 
derived, and the stability of the system is assessed using the movement of the centre 
of mass. 

Chapter 6 uses the fundamental tether models to construct a space-web – a structure 
in space that may be used as a platform to assemble and deploy large space 
structures. 

22 



Chapter 2 

Literature review 

The field of tether research is broad, and spans many branches of engineering, 
physics and mathematics. Tethers may be broadly split into three categories: momentum 
exchange tethers, electrodynamic tethers and structuresbuilt using tethers. 
The notable events in tether history will be critically evaluated. 

While this is not an exhaustive review of tethers, as provided in [Cartmell and 
McKenzie, 2008] or [Cosmo and Lorenzini, 1997], it is intended to cover critically 
the pertinent topics relevant to this thesis. 

2.1 Early pioneers 
In 1895, the father of astronautics, Konstantin Eduardovich Tsiolkovsky visited the 
World’sFair inParis. SeeingtheEiffelTower,he imaginedavast structurestretching 
from the tower to a celestial castle in geostationary orbit above the Earth. From 
this ideaof atower,fundamentallyacompressivestructuredescribed in[Tsiolkovski, 
1959],the ideaofaspaceelevatorwasborn. YuriArtsutanov,Tsiolkovsky’sstudent, 
discussed lowering a structure from orbit to theground in[Artsutanov,1960]. 

23 



Unfortunately, the good work in the USSR was not immediately available in the 
West, and it was a number of years before the idea of large-scale tethers and space 
elevators was published in English. A brief, but informative, article published by 
[Isaacs et al., 1966] appeared in Science,andprovided the impetusforacademics in 
the UK and the Americas to study this new idea. 

In one of the first articles in Acta Astronautica, [Pearson, 1975]identifies the main 
problems(materials limits,vibrationalmodesand stability),and solvesagreat many 
of them from first principles1, introducing tapered tethers and taper ratios. Much 
of the terminology still used derives from that source, for example, characteristic 
height of materials2 . 

Hans Moravec metaphorically severs the link between the space elevator and the 
Earth, postulating the tether as a momentum exchange device in [Moravec, 1977], 
calling it the Skyhook3 . This giant spoke rotates, touching down once and interfacing 
with the surface periodically. He identifies Mars as an ideal candidate for 
the Skyhook, and provides a good grounding for the specific materials and optimal 
layouts that are expanded in further research. 

Several works of fiction have been inspired by these early pioneers, including 
Clarke’s ‘The Fountains of Paradise’. [Clarke, 1979] brings a social aspect to the 
space elevator research that is often overlooked, and is worth mentioning as the 
source in which many young researchers first discover the field. Indeed, Clarke remarked 
in his final interview, see [Das, 2008], that the space elevator will be built 
‘about 10 years after everyone stops laughing’! 

1Onthestate-of-the-artanaloguecomputer in1975,out-poweredby several ordersof magnitude 
by today’s mobile phones! 

2Pearson defines the characteristic height as ‘the height to which a constant-diameter tower 
of the material could be built in a uniform one-g field without exceeding the stress limit of the 
material at the base’. 

3Moravec acknowledges in that paper that the Skyhook idea originated with John McCarthy 
of Stanford. 

24 



2.2 Tether fundamentals 
The foundation textbook, ‘Dynamics of Space Tether Systems’ by Beletsky and 
Levin [Beletsky and Levin, 1993], rigorously introduces the dynamics of tethers, in 
a progressive and pragmatic manner. 

The book discusses the fundamental topics in tether research, such as material 
density, strength, and orbital location, at a level accessible for a newcomer to tethers. 
The book then develops equations of motion for a massless and massive flexible 
tether with variations, covering perturbations and other environmental effects. The 
dynamics are examined in terms of stability and oscillatory behaviour, based on a 
Newtonianderivation. ElectroDynamic(ED)tethers, librationandrotation,deployment 
and retrieval, andLunaranchored and satellitering systems servetocomplete 
the coverage of the book. A very useful set of references is also provided, up to the 
publication year of 1993. 

Achapterof thetextbookbyMcInnes andCartmell[McInnes andCartmell,2006] 
on the dynamics of propellantless propulsion systems covers both solar sails and 
tethers, each sharing the common goal of overcoming the constraints of using a 
reaction mass. This reviews missions involving tethers to date, then moves on to 
summarisetheperformanceexpectationsofhanging, librating,and spinning tethers, 
setting them in the context of results extant in the literature. 

Carroll’s paper (see [Carroll, 1986]) gives an excellent background to the many 
differenttypesoftethermissionproposedandflown, includingtheGemini12 mission 
thatpioneered thetechnique ofgravitygradient stabilisation with theAgena vehicle. 
This paper outlines the various roles of tethers, from the space elevator, librating 
tethers, rotating tethers and momentum exchange tethers. 

25 



2.2.1 Fundamental tether concepts 
Thereareafewfundamental conceptsthat limitthetether, itsperformanceand the 
missions it may usefully provide. 

Rotating tethers(whencomparedtogravitygradientstabilisedtethers) area logical 
progression, and extend the usefulness of the tether system. Tether oscillations 
and other nonlinear dynamics cause concern, and the ever-present possibility 
of severing thetetherduetospacedebris isonlygoing toworsenwithbetteraccess 
to space. Spinning the tether up to rotational speed is a notable problem; in the 
vacuum of space, there is nothing to provide a base to rotate against. 

A tether differs from a rocket engine, a mass-based propulsion technique, in that 
mass isnot ejected. Thisenablestetherstoseverthe linkbetweenpayload andfuel. 
However tethers suffer from a few notable drawbacks: the tether line is vulnerable 
todebrisdamage,tethersare limitedby theirfundamental materialproperties,and 
rotating tethers must maximise their velocity increment. 

Tethers can broadly be split into two categories: the rotating and non-rotating 
tether. Therotatingtether(seeSection2.2.3)isprimarilyconcerned with imparting 
a large velocity increment. The non-rotating tether has a variety of uses, with large 
sections of the literature devoted to control and stabilisation of the tether, as is 
covered in the next section. 

2.2.2 Non-rotating tethers 
Tethers mayprovide stabilisation, control ordamping to a structure or constellation 
in space. Active stabilisation of the tether is usefully demonstrated by ED tethers 
in a highly readable report, [Hoyt, 2002]. Pendulum librations, transverse wave 
oscillations, and skip-rope modes are all investigated and control laws proposed in 
the form of feedback algorithms. 

26 


As Figure 2.1 (taken from [Cosmo and Lorenzini, 1997, p175]) illustrates, the 
rotating tether provides a large advantage over the non-rotating tether in terms of 
the velocity increment to the payload. A frequently cited result is that the apogee 
gains 7L 
in height from a hanging tether of length L, less than 14L 
from a swinging 
tether and more than 14L 
from a rotating tether. 


Figure 2.1: Tether length increment 
L, from [Cosmo and Lorenzini, 1997, p175] 

2.2.3 Momentum exchange 
Momentum exchange is, in general, the main method in which a tether can impart 
a useful velocity increment to the payload. 

Dumbbell tethers are essentially a coupled pendulum system, and have been well 
coveredinthe literatureinthepast,(e.g.withHugen’sclocks,see[Bennettetal., 
2002]). More recently, [Xu et al., 2005] finds that the harmonically excited parametric 
pendulum can be simplified to the Mathieu equation, opening up this area 
to a new branch of analysis. 

Cartmell investigates parametric excitation of the MMET system, finding that 
the motor torque may be minimised while guaranteeing a monotonic spin-up of the 
system. This is achieved by a harmonic modulation of a portion of the payload 
masses, and by manipulation of the radius of gyration of the additional lumped 

27 



masses. Alternatively, given a constant motor torque, the response of the tether 
may be actively chosen by selecting a bifurcation path. 

MisraandModi, in[MisraandModi,1992],provideanexcellentLagrangianformulationfor 
the three-dimensional motion of n-body tethered systems using a multiple-
pendulum modelasthebasisforthe equations. The equations ofmotion are validfor 
large motion and variable lengths, however, the tethers are massless and straight. 
This links the n-body pendulum equations and tethered bodies, and provides an 
excellent template for rigid body motion with many linked tethers. 

Moravecpublished the ideaof themomentumexchangedevice in[Moravec,1977], 
defining the Lunavator4 as a momentum exchange device that touches the surface 
of the moon and picks up, or drops, a payload. 

Thedeploymentcharacteristicsof atethersystemare investigated in[Modi and 
Misra, 1979], with a thorough and in-depth discussion on the derivation of Lagrange’s 
equations of motion for a rotating tether with vibrations. The equations of 
motionfor athree-mass system(two end masses and a massive5 tether) are identified 
as coupled and nonlinear. They study the tether close to the atmosphere, and 
conclude that the out-of-plane motion decays provided the tether is outside Earth’s 
atmosphere. 

Modi andMisra continue their academicpartnership, and report on controlling the 
tether in [Modi et al., 1982], control of tethers used in Space Shuttle-based tethers 
in[Modi etal.,1992]and tethered elevators in[Modi etal.,1993]. 

Perhaps with the Shuttle retiring in 2010, the Shuttle will be proposed less frequentlyasanend-
massforthetether,as in[KyroudisandConway,1988]and[Pascal 
et al., 1999], instead replaced by the upper stage of an Ares rocket. It is interesting 
to note that Pascal mentions ‘crawlers’ – moving payloads along the tether as 

4The Lunavator works best on planetary bodies with no atmosphere, where there is no atmosphere 
to drag against the tether, and on Mars, where the combination of favourable rotation and 
relatively weak gravity suits the system. However, the name Martivator is clumsy, and has not 
caught on. 

5Massive in the sense of a body with a non-zero mass, not massive as in very large. 

28 


an alternative to full deployment. A variation of this moving mass system used to 
control tethers may also be used to control space-webs, as will be discussed later in 
this thesis. 

2.2.4 Motorized momentum exchange tethers 
Torotatethetetherto launch velocity, it isnecessary toprovideameansof rotation. 
On Earth, this is provided by the Earth, however on orbit, one must bring the 
counter-rotation device along with the launch device – in a similar manner to a 
helicopter’s tail boom. 

[Eiden and Cartmell, 2003] briefly summarises the possible role of a European 
roadmap for non-conductive tethers, nominally based on momentum exchange, and 
also for conductive tethers in which electrodynamics, alongside momentum exchange, 
provide propulsion. In the case of the former class small and large payload 
de-orbitareseenasneartermgoals,withfree flying tetheredplatformsand artificial 
gravity systems in the mid-term, followed eventually by spinning tethers providing 
interplanetary propulsion. Gravity gradient stabilisation is an important underpinning 
phenomenon when considering spacecraft stability, and this is particularly the 
casefor long momentumexchangetethers. In[Cartmellet al.,2003],dumbbellmodels 
are considered for momentum exchange tethers, and offshoots and developments 
of this work have shown conclusively that hanging, librating, and spinning tether 
motions are intimately connected to this fundamental phenomenon, as reported in 
[Ziegler and Cartmell, 2001]. Ziegler shows in [Ziegler and Cartmell, 2001] that 
the rotating MMET outperforms a librating tether by two orders of magnitude in 
boosting the apogee of a payload. 

An in-depth treatment of the rigid body dynamics of tethers in space is given 
by [Ziegler, 2003]. In this work the dumbbell tether is modelled at various levels of 
accuracy, and approximate analytical solutions are obtainedby means ofthe method 

29 



of multiple scales6 forperiodic solutions. Comprehensivedynamicalsystems analyses 
are summarisedfordifferent configurations and models, andglobal stability criteria 
for a rigid body dumbbell tether, in both passive and motorised forms, are defined 
and investigated. 

The method of multiple scales is described further in [Cartmell et al., 2003] in 
a review article with several examples. With respect to tethers, the method of 
derivation is covered in great depth. An analytical solution to the MMET system 
is derived using the multiple scales method in [Cartmell et al., 2006], where the 
equations describe a forced and parametrically excited Duffing oscillator. 

Sorensen has authored two interesting papers ([Sorensen, 2003] and [Sorensen, 
2001]) detailing a method of using Momentum eXchange/Electrodynamic Reboost 
(MXER) tethers powered by ED tethers to provide interplanetary travel through 
hyperbolic escape from Earth’s gravity. The second details a boost station that 
sequentially raisesthepayload’s orbitthrough repeated rendezvous and momentum 
exchanges. Of additional note are the appendices detailing analysis techniques and 
mathematical derivations that can be used in tether facility design. 

With any rotating system,thepossibility of rotational instability arises. [Valverde 
and vanderHeijden,2008]find that amagneticbuckling of awelded rod isfound to 
be described by a surprisingly degenerate bifurcation. This has serious implications 
for using ED tethers as well as static (conducting and non-conducting) deployed 
tethers for rotational launch systems. 

[van der Heijden, 1995] covers the effects of bearing clearance driving a nonlinearly 
coupled driven oscillator in rotordynamics, with close parallels to the MMET 
system. Partial mode-locking then leadstoquasi-periodicmotionof therotors,and 
perturbations of the system lead to more complicated motions. 

6Themultiplescalesmethod waspopularised in[Nayfeh andMook,1979]. 

30 



2.2.5 Staging with momentum exchange 
As the fundamental material limits of tethers limit the useful 
V 
that may be 
imparted, novel techniques such as staging tethers are explored. 

A short and digestible article by Forward and Hoyt in Scientific American (see 
[Forward and Hoyt, 1999]) first showed the ingenious concept behind the use of 
multiple, staged tethers to increase velocity without the necessity of extreme design, 
for Earth-Moon payload transfer. [Forward, 1991] continues this work by pairing 
theLunavator with aLowEarthOrbit(LEO)tether. 

Theconceptof staging isexpandedfurther in[McInnes andCartmell,2006] and 
[Cartmell et al., 2004]. 

Following upfromtheground-testsof amomentum exchangetetherperformed on 
ice, [Cartmell et al., 2003] takes this a stage further, and rigorously examines the 
release and de-spin of payloads. 

Large-scaleelectrodynamicandmomentumtransferbetweenplanetswasstudiedin 
theNASAMXERprogram,by Forward,Hoyt,Sorensen and others. Initial concept 
design work in [Sorensen, 2001] outlined a massive tether system to send 20 - 80 
tonnes to Lunar or Mars orbit, with a tether length of up to 100km. Sorensen 
finds that hyperbolic injection is possible in [Sorensen, 2003], and this opens up 
the MXER system to send payloads to Mars. It must be stated that the necessary 
velocity to launch a payload to Mars is not attainable with current materials7, to 
do so would require a significant increase in the specific strength of the tether. 

2.2.6 Deploying and recovery of tethers 
Further work on using tetherbased transfers was reportedby[Lorenzini et al.,2000] 
in their landmark paper in which staged tethers in resonant orbits are proposed 

7The
V 
forLEOtoLMO(LowMarsOrbit)isapproximately6.1km/s 
andthe maximum characteristic 
velocity of a single strand tether is approximately 2.7km/s. See Appendix D, Table D.1 
for current materialproperties. 

31 


as being more mass efficient than single tether systems8 . Lorenzini et al. briefly 
refertotether orbit raising results citedby[Carroll,1986] for radial separation as a 
function of tether length, and conclude that spinning staged tethers could provide 
an ideal transferrateof fivetransfersperyear. Thetransferrateof astaged system 
is determined by the periodic realignment of the apsidal lines of the two stages, 
whereas in the case of a single tether it is dependent on the time required for re-
boosting the stage. 

Continuing with the theme of propulsion of a small payload tethered to a large 
mass intheformof aspacestationor largeShuttle,[Pascal et al.,1999]investigated 
the laws of deployment and retrieval by means of a three dimensional rigid body 
model of adumbbelltether inboth circularand elliptical orbits,expanded in[Pascal 
et al., 2001]. Several laws are proposed and analytical solutions for small planar 
and non-planar motions of the tether are given, showing that equilibrium tension 
can be stated as a function of instantaneous tether length and corresponding axial 
acceleration,forwhich controllawscanbestipulated. It isshownthatdeployment is 
generallystablewhereasretrieval isnot. Various lawsareexaminedfordeployments 
and retrievals, and also for crawler configurations in which the end payload moves 
out along a pre-deployed tether and how this can mitigate the inherent instability 
of retrieval. 

The next conceptual step to take when considering deployment is to include some 
form of flexibility within the tether, and an interesting study of this was published 
by[Danilinetal.,1999], inwhichtheelastictethermodel of[NoandCochranJr., 
1995]isusedbut withdifferent variablesandderivation. Theobjectiveof thispaper 
was to consider deployment of a completely flexible tether from a rigid rotating 
space vehicle under the influence of a central gravitational field. The tether is 
modelled using a Newtonian derivation as a series of discrete masses interconnected 
by masslesselementsand with internal viscousdamping. Theauthorsmakethevery 

8Theypropose using afacility topayload mass ratio of1 :3 using current materials. 

32 


important point that tether element forces cannot be compressive, so conditions 
within the numerical solution algorithm have to be set up to accommodate the 
consequential folding effects. Two numerical examples are summarised; one for a 
swinging terrestrial cable with an end mass, which starts from a horizontal initial 
condition, mainly as a verification of the model in those conditions; and the other 
for plane motion of a space vehicle deploying a relatively short 3km 
tether, with 
elemental spacing of 100m. The deployment is linear and conditions are set up to 
apply smooth braking of the tether to a halt at the end of the deployment. 

2.2.7 Capture and rendezvous of tethers 
An implicit assumption with tether staging is that the payload will, at some point, 
have to rendezvous with a second system. [Lorenzini, 2004] provides an in-depth 
treatment of a spinning tether loop with an extended time opportunity for errortolerantpayload 
capture withinhigh
V 
propulsiontoGTOandEarth escape orbit. 
Theconfiguration issuch thattheendsof the loop arefurthest awayfromthecentre 
of mass, where the loop is at its narrowest. The concept is simple in principle. The 
satellite contains a harpoon which shoots through the loop and hooks onto the loop 
to capture the satellite. This makes it tolerant of large longitudinal position errors 
and reasonable lateral errors as well as some out-of-plane error. The capture is soft, 
and so caters for some velocity mismatch. 

[Williams et al., 2003] outlines a ‘momentum-enhanced gravity-assist’ technique 
to capture a payload on a previously hyperbolic orbit into planetary orbit. This is 
feasible in theory, however requires accurate control and a large tether deployment 
rate, which may limit the practical usefulness of the technique. 

A subsequentpaper,[Williams et al.,2005], systematicallydescribestheproblems 
in payload capture (spatial and temporal matching, post-capture dynamics and 
failure concerns), and designs a suitable rendezvous and control system involving a 

33 



crawler to successfully capture a payload. 

2.2.8 WSB trajectories 
Perhapsoneof themostinteresting concepts intrajectory analysis,atleast asfaras 
thetether isconcerned, istheWeakStabilityBoundary(WSB) trajectoryproposed 
by [Belbruno, 1987]. This technique allows the majority of the 
V 
for an Earth-
Moon mission to be concentrated in the first burn, with a relatively minor capture 
burn at the end of trajectory. 

If this is compared to the 
V 
requirement of the Hohmann two-burn technique9 , 
it becomes clear that a tether-based launcher system could provide a more efficient 
route to the Moon. A thoroughly good overview of the WSB method is given in 
[Ross,2004,p112],where itisstated thatthefuel required isloweredby about20%. 
As is common with large research groups, there are many valuable papers on WSB 
transfers and the Interplanetary Transport Network from the CalTech researchers 
that are listed in Ross’ thesis. 

Apractical exampleof theWSB method isexplained clearly in[Koonet al.,1999], 
giving an example of a trajectory similar to the Hiten Lunar probe. The WSB 
trajectory10 has since been used to successfully launch NASA’s Genesis mission to 
study the solar wind(see[Lo andRoss,2001]). 

TheWSBtrajectoriesareanumbrellaforamultitudeofpossibleroutes, incontrast 
to the simplicity of the Hohmann or bi-elliptic trajectories. A useful geometric 
method of finding such a WSB trajectory is outlined in [G´omez et al., 1993], taking 
thedestinationastheEarth-SunL1point. Oncethispoint isreached,thepathways 
are open to a great many places in the Solar system. 

9Hohmann’sseminalbookon ‘TheAttainability ofHeavenlyBodies’maybefoundin itsoriginal 
form[Hohmann,1925](inGerman) ortranslated toEnglish in[Hohmann,1960] 

The WSB trajectory is hard to visualise and perhaps 2D plots of the spacecraft’s path in 
inertial and rotating space are difficult to extrapolate mentally to 3D space. Thankfully Lo 
has made videos of the spacecraft path on his website: http://www.gg.caltech.edu/~mwl/ 
communications/communications2.htm 


34 


A method of using a MMET with a WSB trajectory is outlined in [McKenzie and 
Cartmell, 2004]. A small satellite MMET system, under 1000kg, is able to launch 
a 10kg 
micro-satellite from MEO to Lunar capture orbit in around 100 days. This 
system may be recovered and re-launched every month until the useful life of the 
satellite is reached. This relies on a very small safety factor of 1.3 in the main 
tethers, and does not account for the additional stator arm in the MMET, however 
this shows that the current state of technology is almost ready to sustain an orbital 
tether launch system. 

2.3 Tether derived structures 
In the same way that orbital tethers have evolved from the space elevator, space 
structures may be constructed from tethers. 

2.3.1 Deployed space structures 
In1968,PeterGlaser suggested that a solar collector11 couldbeplacedingeostationary 
orbit [Glaser, 1968]. Operating in high Earth orbit, this would use microwave 
power transmission to beam solar power to a very large antenna on Earth where it 
can be used in place of conventional power sources. The advantages to placing the 
solar collectors in space are the unobstructed view of the Sun, the fact that it is 
then unaffected by the day/night cycle, weather, or seasons. For efficient operation, 
the satellite antenna must be between 1 and 1.5km 
in diameter and the ground 
rectenna around 14km 
by 10km, generating between 5 and 10GW 
of power. One 
option of constructing this huge collector is manufacturing the cells on Earth, and 
constructing them in space with the aid of robots. 

An alternative approachtothe on-orbit assembly ofsuch a massive structure would 
betheuseofwebs,as in[Kayaet al.,2004a]and[Kayaet al.,2004b].Alargegeneric 

11That is, a satellite for collecting solar power. 

35 


net-likestructurecould firstlybedeployed inorbit. Oncethenet isstabilised,robots 
could be used to crawl over the net towards specified locations and move solar cells 
into the desired positions. This enables the construction of such a satellite using 
robot assistance to be faster and cheaper12 . 

Other largespacestructureshavebeenproposed that wouldmakeuseof ageneric 
reconfigurable platform. In the case of the orbiting stellar interferometer, [DeCou, 
1989] showed that planar deformation of a spinning system comprising three collimating 
telescopes at the corners of an equilateral triangle made up from three interconnecting 
tethers would be inevitable due to the inertia of the tethers. Clearly 
inertia-less tethers will not deform centripetally and will, instead, merely stretch 
intostraight linesduetothetensioncreated along their lengthby thecornermasses 
as the whole system rotates. Perhaps the addition of a control system would make 
this particular structure a more realisable option. 

This directly led to the successful deployment of the Inflatable Antenna Experiment(
IAE),andled atrancheof successfullydeployed structuresinspace, including 
the ESA Cluster mission as detailed in [Andi´on and Pascual, 2001]. 

Studies have previously shown that robots may be deployed in this way to reconfigure 
the net structure [Kaya et al., 2004a], [Nakano et al., 2005]. A international 
collaboration from the University of Glasgow, KTH Royal Institute of Technology 
in Sweden and the ESA plans to test the concept on a sounding rocket in 2010, as 
reported in a [University of Glasgow Press Release, 2009]. 

2.3.2 Space-webs 
If the generic satellite is to be used as an element of infrastructure, it is desirable 
to start studying the dynamics and deployment of such a structure. A type of 

12To a degree, the question of whether a method of power generation is ‘better’ is largely a 
political decision. 

36 


satellite, called ‘Furoshiki’13 is outlined in [Nakasuka et al., 2001], with an overview 
of the concept, a simulation of a membrane satellite and an evaluation of the control 
strategy needed to stabilise such a satellite. 

Further work on the deployment of a square membrane is examined in [Tibert 
and G¨ardsback, 2006], where a membrane is attached to four daughter satellites 
and deployed while the satellite is both free and rotating. It is shown that the free 
deployment without damping is not possible, however a torque can be applied to 
the central hub to stabilise the deployment. 

Once the membrane is deployed, robots such as reported in [Kaya et al., 2004a] 
may be tasked to assemble or reconfigure the structure. 

A multi-tethered structure containing n 
masses is investigated in [Pizarro-Chong 
andMisra,2008],for an openhub, closedhub(wheel) anddoublepyramidformations. 
A system of Lagrange’s equations of motion is constructed and the stability 
of the three configurations were tested in a variety of configurations – parallel and 
perpendicular to the orbital plane and with and without an initial spin-rate. The 
dual-pyramid formation was found to be the stable in most tests, with all configurations 
benefiting from a rotating platform. 

A similar analysis in [Wong and Misra, 2008], involving a multi-tethered structure 
near the Sun-Earth L2 point, examines the planar dynamics of the satellite. Lagrange’s 
equations are used to model the system, with a linear feedback controller to 
control thetether lengthsand angulardisplacementsof thesystem. Theend masses 
were successfully controlled, wielding a spiral pattern which allows the satellite to 
be used as an optical interferometer. 

13After the traditional Japanese wrapping cloth. 

37 


Chapter 3 

Tether modelling 

The steps to construct a mathematical model of a tether orbiting a planet are 
outlined in this chapter. Starting with a simple model of a point mass orbiting a 
massivebody,each stagebuildsonthe lastandprogressivelyadetailed(and more 
importantly, validated) model will be outlined. 

3.1 
Constructingtheequationsof motionof asystem 
Amethod todescribeasystem intermsofOrdinary DifferentialEquations(ODE) 
is outlined below. For each mass point of the system: 

. 
the position of each mass point is described in local coordinate space 

. 
theCentreofMass(CoM) of thesystem isfound 

. 
the position is translated and rotated1 into inertial coordinate space, using 
matrix and vector operations 

1The centre of mass effectively links the two coordinate systems by providing a common point 
of reference between the two. 

38 


then, for the system as a unit: 

. 
the Lagrangian is found by expressing the total energy of the system 

. 
generalised coordinates are chosen to suit the system. 

. 
velocities of each mass point are derived 

. 
Lagrange’s equations are constructed 

. 
Lagrange’s equations may be numerically integrated using suitable initial conditions 


The remainder of this chapter will fully outline the steps needed to construct and 
validate ageneral system of equations,focussing specifically on constructing aplanar 
dumbbell tether on a circular equatorial Earth-centred orbit. 

3.1.1 Validating code modules 
It isessentialaspart ofthescientificmethodtoset up anexperiment,or inthiscasea 
numerically solved mathematical model, that is both repeatable and verifiable. To 
this end, a method of setting up equations describing each component’s position 
using repeatable and testable modules is introduced here. 

It is of fundamental importance, when building a mathematical model, to ensure 
that the model is verified and validated insofar as practically possible. As such, 
a method of constructing a system of validated modules is introduced to verify 
that, for example, a module to rotate a vector about an axis does this accurately 
and repeatably. Therefore, for each component of the system, a testable module is 
produced and the system as a whole is much easier to validate as the sum of its 
parts. 

In Mathematica. 
, modules of code may be configured and tested such that a 

39 



one-line testable routine may be evaluated. Assigning: 

.. 


10 0 

.. 


.. 


..

XRotation[]= 0 cos() -sin() (3.1) 

.. 


.. 


0 sin() cos() 

it isthenpossibletoseparateand logicallyevaluatetheright and lefthand sidesof 
Equation 3.1 to see if they are identically equal. If the right hand side is identically 
equal to the left hand side, the code is validated, if not, the code fails validation. 

In this way, it is possible to establish a repeatable and valid technique for constructing 
the equations of motion of the system. 

3.2 Centre of mass modelling 
3.2.1 Mass points & body selection 
It is essential when using the Lagrangian method that the movement of the bodies 
concerned are described accurately. The modelling tasks can be broadly separated 
into expressions concerning: 

. 
bodies containing mass which contribute to the kinetic and potential energies 

. 
external or internal conservative forces that may contribute to the energy sum 

(e.g. springenergies) 
. 
external or internal forces that may contribute to the non-conservative right-
hand-side terms in Lagrange’s equations 

As such, the principles of system modelling will be demonstrated using a simple 
exampleofthecoplanardumbbelltether incircularEarth orbit,shown inFigure3.1. 

40 



 
















































Figure 3.1: Symmetrical dumbbell tether 

Central to the system, the facility mass houses the entire structure of the system; 
the power generation, communications equipment, etc.. This is expected to form 
the majority of the mass of the system. 

Thetwopayloads arethe raison d’ˆetre of the momentum exchange tether system. 
They are assumed to have mass and are rigidly connected to the facility by the 
tethers and are usually a significantly smaller fraction of the system mass. 

The two tethers are assumed to be massless. This is usually a significant assumption 
to make because the fractional mass of a 100km 
tether is non-trivial. In this 
case, itisareasonableassumptiontodemonstratetheprinciplesof tethermodelling 
without making the mathematics overly arduous. 

Thetethersholdthepayloadsrigidlyand symmetrically inplaneaboutthefacility 
mass, ensuring that the tethers are colinear about one common axis through the 
centre. 

Theotherbody of note is,of course,theEarth which acts indirectlyonthissystem 
providing the gravitational potential energy. 

3.2.2 Centre of mass calculations 
The centre of mass can have a considerable effect on the motion of this system. 
Consider an asymmetrical payload release, for example, when only one payload is 
released instead of both, as normally occurs. The centre of mass would undergo an 
instantaneous step-change and, consequently, the orbit of the system would change. 

Assuming the dumbbell is in a circular orbit aligned along the vertical (gravity 
gradient) direction, if the uppermost payload is released the excess kinetic energy 

41 



of the orbit will cause the payload to describe an ellipse with the separation point 
as the perigee. The remainder of the system2 will tend to de-orbit as the potential 
energy of the system is lower than needed to sustain a circular orbit. This separation 
point becomes the apogee of the new system’s orbit, as shown in Figure 3.2. 






























Earth 
Figure3.2: Symmetricaldumbbelltether on a circular orbit(solid). Afterpayload 
release,thefacility systempath at apogee(dash) andpayload atperigee(dots) 

Inasymmetrical system,thecentreof mass is located coincidentonthefacility. If 
thesystem isasymmetrical,thenthecentreof masswillneed tobecalculated. This 
is highly important to the accuracy of the model and in most cases necessary to 
integrate the equations of motion of the system. Most of the problems encountered 
whennumerically integratingtheequationsof motion inMathematicaweresolvedby 
inserting a mathematical expression that accuratelydescribes the(variable)position 
of the centre of mass in the positional equations which then feed into Lagrange’s 
equations. 

The Centre of Mass (CoM) calculations give the location of the CoM from an 
arbitrarily chosen point. In this case it is convenient to chose the facility mass as 
theoriginpoint(and thecentreof a local coordinatesystem) and refertoallother 

2That is, the lower section of the system comprising the facility, two tethers and one payload. 

42 



points in the tether system relative to the facility location. 

This may be alternatively expressed as constructing a local coordinate system 
{Xlocal,Ylocal,Zlocal}centred on the facility mass, in this case, aligning the X-axis 
with the radius vector. 

The position of the centre of mass about the facility for n 
masses each with positions 
{Xi,Yi,Zi}, in terms of the local {X, 
Y, 
Z}coordinate system centred on the 
facility mass is: 

. 
. 
n 
n 
n 
. 
. 
. 
. 
. 
MiXi 
MiYi 
MiZi 
. 
. 
. 
. 
i=1 i=1 i=1 Pfacility.CoM = , 
, 
(3.2) n 
n 
n 
. 
. 
. 
. 
. 
. 
Mi 
Mi 
Mi 
. 
. 
. 
i=1 i=1 i=1 

Once the CoM relative to the facility mass has been found, a vector describing the 
position of any mass point in the inertial frame of reference3 may be constructed. 
This is used in later equations by taking the position from the centre of mass to 
the facility mass: 
PCoM.facility = -Pfacility.CoM (3.3) 

3.2.3 Positions of mass points 
A technique to obtain the location of each mass point relative to the facility is 
outlined below, using a 2D dumbbell system as an illustrative example. 

To assemble the equations of motion, a series of rotations and translations are 
performed on the initial vectors to transform the local space coordinates into inertial 
space coordinates. The initial vectors are chosen in synchronisation with the 
generalised coordinates and the rotation angles. 

Figure3.3showstheorbital motion,constrained tothe inertialX-Yplaneand the 
local X-Y plane, of an asymmetric dumbbell tether. Note that the local X-axis is 

3In the Earth centred system. 

43 


Earth 
LYlocalXinertialXlocalCoMRadius=P0!CoM

YinertialjPCoM!facilityEarth 
LYlocalXinertialXlocalCoMRadius=P0!CoM

YinertialjPCoM!facility
Figure 3.3: Asymmetrical dumbbell in inertial axes 

parallel to the radius vector. 

The inertialpositionvectorPinertial of each masspoint, j, isassembledfromthevector 
sum of the component vectors, taking into account the translation and rotation 
of each component. 

Pjinertial = R,Z · P0.CoM -Pfacility.CoM + R j 
,Z · Lj 
(3.4) 

Where P0.CoM = {R, 
0, 
0}, and represents the vector of the centre of mass from 
Earth4 . The rotation matrices and matrix notation R j 
,X are explained in Section 
3.5. 

The five stages in constructing the position of the mass point Pj 
are shown in 
Figures 3.4, 3.5, 3.6, 3.7 and 3.8, and the position vectors for an arbitrary mass 
point are given in Equations 3.5, 3.6, 3.7, 3.8, 3.9 and 3.10. 

Figure 3.4 shows the mass point aligned along the X-axis of the local coordinate 
system, in this case using a payload mass with length L 
as an illustrative example. 

4i.e. the Radius vector. 

44 


Ylocal 


LPjXlocalfacility
Figure 3.4: Local coordinate construction 

Ylocal 


L
 
PjXlocalfacility
Figure 3.5: Local coordinate construction after rotation through angle . 
about the 
facility mass 

45 



YlocalXlocalPj
 
LPCoM.facilityfacilityCoMYlocalXlocalPj
 
LPCoM.facilityfacilityCoM
Figure 3.6: Local coordinate construction after translation from the facility to the 
CoM 

Earth 
YlocalXlocalPj
 
LPCoM!facilityfacilityCoMYinertialXinertialR
Figure 3.7: Local coordinate construction after translation from the CoM to the 
inertial coordinate system 

46 



Earth 
YinertialYlocalXlocalPj
 
LPCoM!facilityfacilityRCoM

XinertialEarth 
YinertialYlocalXlocalPj
 
LPCoM!facilityfacilityRCoM

Xinertial
Figure3.8: Local coordinate construction after rotation through . 
about the inertial 
coordinate system 

Rotating this through an angle of . 
about the facility as shown in Figure 3.5. This 
is translated to align the origin with the centre of mass as shown in Figure 3.6. 
A further translation is performed through the inertial X-axis about a distance R, 
giving the origin as the inertial coordinate system, as shown in Figure 3.7. Finally, 
the inertial coordinatesystem isrotated abouttheoriginaround anangle . 
as shown 
in Figure 3.8. 

47 



The vectors of each are shown below: 

.. 


L 


.. 


.. 


..

P1 = 0 (3.5) 

.. 


.. 


0 

.. 


L 
cos( ) 

.. 


.. 


..

P2 = R ,Z · P1 = L 
sin( ) (3.6) 

.. 


.. 


0 

.. 


L 
cos( ) -CoMX 

.. 


.. 


..

P3 = P2 -PCoM.facility = L 
sin( ) -CoMY (3.7) 

.. 


.. 


-CoMZ 

.. 


L 
cos( ) -CoMX +R 


.. 


.. 


..

P4 = P3 + R 
= L 
sin( ) -CoMY (3.8) 

.. 


.. 


-CoMZ 

P5 = R,Z · P4 = 

.. 


cos. 
(L 
cos( )-CoMX + R)-sin. 
(L 
sin( )-CoMY) 

.. 


.. 


..

sin. 
(L 
cos( )-CoMX + R)+cos . 
(L 
sin( )-CoMY) (3.9) 

.. 


.. 


-CoMZ 

P5 = R,Z · P0.CoM -Pfacility.CoM +(R ,Z · P1) (3.10) 

The process is repeated for each mass point under consideration. 

3.3 Constructing Lagrange’s equations 
Lagrangian mechanics is primarily concerned with the trajectory of an object, derivedbyfinding 
thepath which minimisestheaction,aquantity which isthe integral 

48 



of the Lagrangian over time. As this is an energy based approach, the process of 
evaluating the kinetic and potential energy of the system will be outlined and the 
choice of generalised coordinates will be explained. 

3.3.1 Energy modelling 
For every mass point in the system with position in the inertial frame, Pj, the 
following steps are undertaken in order to find the Lagrangian energy expression L: 

1. the respective velocities, Vj 
are found: 
. 


Vj 
= Pj 
(3.11) 

@t 


2. thekinetic energies(linear Tlin = 21 mv2 and rotational Trot = 21I!2) are obtained 
and summed: 
n 
1 

Tlin = mj 
Vj 
· Vj 
(3.12) 

2 

j=1 
n 
1 

Trot = mj 
Pilocal · Pjlocal !j 
· !j 
(3.13) 

2 

j=1 

where the vector of angular velocity, !j 
is the first derivative of the angular 
position of each point, {j,j, j 
} 

3. the potential energies, U, are obtained and summed: 
nn 


µmj 
µmj

U 
= 
. 
= 
. 
(3.14) 

Pj 
Pj 
· Pj
j=1 j=1 

where µ 
is the standard gravitational parameter. For the Earth, the value is 
µ 
=3.986* 1014 m3/s2 . 

4. and the Lagrangian is found: 
L 
= Trot + Tlin -U 
(3.15) 
49 


For each mass point, the Lagrangian energy expression is constructed by considering 
the total kinetic and potential energies of the system. 

d 
@T 
@T 
@U 


- += Qj 
(3.16) 

dt 
@q.j 
@qj 
@qj 


Lagrange’s equations are generated for all the generalised coordinates as specified 
in Equation 3.16. 

The forces are then used to calculate the right hand side of Lagrange’s equations, 
Qj 
, through consideration of the virtual work. 

Primarily, the main non-conservative force acting on the tether system will be the 
motortorque. Thisconcept iscoveredfurther inZiegler’sPh.D.thesis[Ziegler,2003, 
p43-45]. 

3.3.2 Choosing generalised coordinate systems 
Thegeneralised coordinatesused inderivingLagrange’sequations inSection3.3are 
chosen to be independent coordinates. These coordinates need not be orthogonal 
nor Cartesian, in fact it is limiting to assume they are either, however they must 
describe an independent degree of freedom of the system. 

In the case of the orbital variables R 
and , these are chosen in preference to 
the inertial variables Xinertial and Yinertial to enableperiodic analysis overthe orbital 
cycle. 

Inthe localcoordinatesystem,thechoiceof ageneralised coordinate is lessstraightforward. 
In a two generalised coordinate system with an in-plane and out-of-plane 
motion, there are many angles that could be chosen as the generalised coordinate, 
andthisposes aproblem over choice of variables. The equations of motion(and 
therefore the corresponding set of Lagrange’s equations) that describe one system 
of rotations would be different from the others, but the motion would have to be 

50 



identical when comparing the many possible rotations systems. This implies that 
for the motion to be identical5 there could be many possible systems of equations 
that could, when integrated, provide the motion of the system, and that there may 
be one or more systems of equations that provide a more elegant solution than the 
rest. Two systems of rotations are investigated in Section 3.5. 

The goal in selecting a “good” choice of generalised coordinates is that they are 
easily interpreted in graphical form, and the equations generated are compact and 
meaningful to interpret. 

3.4 Non-conservative forces 
Any non-conservative forces that appear in the system need to be included in the 
equations of motion in terms of virtual work. These are contained within Ziegler’s 
Ph.D. Thesis [Ziegler, 2003, p43-45], and will be contextualised here. 

The non-conservative force is represented by Qj 
and can be evaluated by considering 
the virtual work done by the motor torque summed over each mass: 

@Pmass 

Qj 
= F 
· (3.17) 

@qj

mass 

with j 
representing each of the n 
chosen generalised co-ordinates and Pmass as the 
position vector of the mass under consideration. F 
is the non-conservative force. 

The3×3rotationmatrixforageneralised rotation inthetetherbody axes6 (as will 
be explained in Section 3.5) is multiplied by the length vector to give the position 

5If the systems were non-identical then, generally speaking, there would either be flaw when 
choosing the generalised coordinates or a mistake in the equations. The process of simulating the 
motion of a system should be entirely independent of the means used to arrive at the end result! 

6Rotations are only provided here for the Y-and Z-axis. 

51 



of each mass in the tether body axes : 

.. 


Lmass 

. 
. 
. 
. 
Pmass = R,Y · R ,Z · . 
0 . 
(3.18) 
. 
. 
. 
. 
0 

The resultant 3 × 1 vector is differentiated with respect to each of the n 
chosen 
generalisedco-ordinates togive an n×3matrix. Theforcevectorexpressed intether 
body axes isthenfoundby multiplying thegeneralised rotationmatrixby theforce 
vector: 
.. 


0 

.. 


.. 


..

Force = R
,X · R,Y · R ,Z · 0 (3.19) 

.. 


.. 


/Lmass 

where t 
is the non-conservative force converted into the rotating frame. 

Thedotproduct of the3×1 force vector and the n 
×3 matrixthengives an n 
×1 
vector containing the Qj 
terms for the n 
chosen generalised co-ordinates to be used 
in the right hand side of the equations of motion. 

3.5 Rotation systems 
A comparison will now be made between two rotation systems that specify the 
position of an arbitrary mass point in the local axes system. 

As discussed earlier7 there are multiple ways to specify the position of the mass. 
No one system may be termed the definite solution,however, one may bepreferable 
to choose if it generates a more easily interpreted system of equations. 

7See section 3.3.2. 

52 



3.5.1 Rotation summary 
Three rotation matrices are set up to describe the direction of a single 3D vector 
about its origin using a series of rotations about the local X-, Y-and Z-axes. 

As the three rotations can rotate an arbitrary vector around the origin, the initial 
vector must be defined. For the rotation systems that follow, a vector aligned with 
the X-axis in the positive direction starting at the origin has been chosen. The 
series of rotations are then performed sequentially to give 3 new coordinate axes 
(two intermediate andone final) for 3 rotations. 

The Rotation R,B is defined here as a rotation matrix acting about the B-axis 
and rotating about an angle . 

The new vector is found by rotating the starting vector coordinates by means of 
three rotation matrices about the local origin. The rotation system will be one of 
the six permutations from the list of three possible rotations: R,X, R,Y and R ,Z 
given in Equations 3.20, 3.21 and 3.22. 

Therotationsystemthatrotatesthepoint first around theX-axis,thentheY-axis, 
then the Z-axis is termed the XYZ rotation system. 

.. 


10 0 

.. 


.. 


..

R,X = 0 cos() -sin() (3.20) 

.. 


.. 


0 sin() cos() 

.. 


cos() 0 sin() 

.. 


.. 


..

R,Y = 
. 
0 10 
. 
(3.21) 

.. 


-sin() 0 cos() 

.. 


cos( ) -sin( )0 

.. 


.. 


..

R ,Z = sin( ) cos( ) 0 (3.22) 

.. 


.. 


0 01 

53 



The order of matrix multiplication is important; for the XYZ rotation sequence, 
the vector multiplication is evaluated sequentially from the right hand side: 

R ,Z · R,Y · R,X · Porig. 

= R ,Z · R,Y · R,X · Porig. 

Notewell,thisdiffersfromtheZYX rotationsystem intheorderof rotations. The 
ZYX rotation system first rotates the vector by the Z-axis, then the Y-axis, then 
the X-axis. 

R,X · R,Y · R ,Z · Porig. 

= R,X · R,Y · R ,Z · Porig. 

When looking at the matrix rotation equation, this may seem counter-intuitive to 
label the rotational sequence from right to left8 . However, it reflects the sequence 
of rotations, even if it is backwards from a point of view of reading the matrix 
multiplication subscripts. 

Using thethreerotationsgivenhere inthethreeaxes,thereare6possiblecombinations 
of defining a rotational sequence, as shown in Table 3.1. These describe the 
sequence of rotations when rotating a point around the origin in 3D and the three 
possible rotations in 2D that may be derived from the 3D rotation. 

The 2D rotations are not unique to a single 3D rotation. For example there are 
three XY rotations that may be made in 2D that correspond to XYZ, XZY or 
ZXY rotations in 3D space. 

Two of these sequences will be investigated using a simple 2D rotation with the 

8Indeed,this isaratherarbitrarydecision,but itcouldbearguedthatthisfollowsthearbitrary 
direction of text in-keeping with the custom of languages of the Western hemisphere. In choosing 
the nomenclature, it makes sense to make XYZ define the rotational sequence rather than the 
order in which the matrices are multiplied purely because most non-specialists would grasp the 
concept of order of rotations before learning three-dimensional matrix algebra! 

54 


Order of rotation in . . . 
3D 2D 
R, 
R, 
R , 
1 2 3 
X Y Z XY XZ YZ 
X Z Y XZ XY ZY 
Y X Z YX YZ XZ 
Y Z X YZ YX ZX 
Z X Y ZX ZY XY 
Z Y X ZY ZX YX 

Table 3.1: Order of rotations in 3D and 2D 

sequenceof rotationsabouttheYZ-andZY-axes, inSection3.5.2andSection3.5.3 
respectively. 

3.5.2 YZ Rotations 
The YZ series of rotations9 rotates a point lying on the X-axis around first the Y-
axis, then the Z-axis. The X-axis rotation is not included in this system for ease of 
visualisation. 

Rotating the original vector Porig. = {L, 
0, 
0}T 
, shown in Figure 3.9a about the 
Y-axis through an angle of a 
gives the following, as shown in Figure 3.9b: 

.. 


L 
cos() 

. 
. 
Pinter-YZ = R,Y · Porig. = . 
. 
. 
0 . 
. 
. 
(3.23) 
. 
. 


-L 
sin() 
Rotating this intermediatestagearound theZ-axisthrough anangleof . 
gives the 

9Or to be specific the XYZ series of rotations in 3D space when the X rotation is zero. 

55 



following, as shown in Figures 3.9c and 3.9d: 

. 
. 
L 
cos()cos( ) 
. 
. 
. 
. 
Pfinal-YZ = R ,Z · Pinter-YZ = . 
. 
L 
cos()sin( ) . 
. 
(3.24) 
. 
. 


-L 
sin() 

A diagram of the complete rotation sequence is are shown in Figure 3.9e. 

Theout-of-planeanglegivenby theYZ seriesof rotations istheangle . However, 
the in-plane angle is not the angle  . The in-plane angle lies in the plane normal 
to the ZYZ plane, and intersects the XZ plane. 

The in-planeangle(givenby the XY-XYZ plane)isprojectedontotheXY-inertial 
plane to give the angle  . 

3.5.3 ZY Rotations 
TheZY systemof rotations isgeneratedby rotating thestarting vectoraligned with 
the X-axis around first the Z-axis, then the Y-axis. This can be thought of as the 
ZYX series of rotations with a zero X-axis rotation – therefore giving only a ZY 
rotation. 

Rotating the original vector Porig. = {L, 
0, 
0}T 
, shown in Figure 3.10a about the 
Z-axis through an angle of . 
gives the following, as shown in Figure 3.10b: 

.. 


L 
cos( ) 

.. 


.. 


..

Pinter-ZY = R ,Z · Porig. = L 
sin( ) (3.25) 

.. 


.. 


0 

Rotating this intermediate stage around the Y-axis through an angle of a 
gives 

56 



ZZ Z 

Y 
y 
a 
a 
(a)Rotations YZ – start 


a 
Y 
a 
a 
ZYXY
(b)Rotations YZ – 


X 

ZZ 

a 
Y 
y 
a 
a 
XZXYZXYZY(c)Rotations YZ –  
XX 

(d)Rotations YZ – a 
then . 


y 
a 
Y 
y 
a 
a 
YZZYZXZXYZXYZY
Z 

a 
Y 
y 
a 
a 
ZYZYZXYZZYXY(e)Rotations YZ – full rotation shown 
Figure 3.9: Rotation sequence for YZ rotation: Rotating about Y-axis then  
about Z-axis. is shown as a negative rotation here for ease of visualisation. 
57 
X 



ZZ 


Y 
h 
y 
(a)Rotations ZY – start 


Y 

XZ 


XX 

h 
y 
(b)Rotations ZY –  


a 
Y 
h 
y 
XZYZY
(c)Rotations ZY – 


ZZ 

XZ 


XZ 


XX 

a 
Y 
a 
ZYXZYXY
(d)Rotations ZY –  then 


Figure 3.10: Rotation sequence for ZY rotation.. Rotating . 
about Z-axis then a 
about Y-axis. a 
is shown as a negative rotation here for ease of visualisation. 

the following, as shown in Figures 3.10c and 3.10d: 

.. 


L 
cos( )cos() 

. 
. 
. 
. 
Pfinal-ZY = R,Y · Pinter-ZY = . 
. 
L 
sin( ) . 
. 
(3.26) 
. 
. 


-L 
cos( )sin() 

A diagram of the complete rotation sequence is shown in Figure 3.10d. 

Conversely to the YZ series of rotations, the out-of-plane angle given by the ZY 
series of rotations is not the angle . However, the in-plane angle is now the angle 
 . The out-of-plane angle is given by the angle in the XZY -XZ plane between the 
fully rotated vector and the X0 -Y0 inertial plane. 

The out-of-plane angle(givenby the XZY -XZ plane) is projected onto the XY-
inertial plane to give the angle . 

58 


The out-of-plane angle, , may be found by simple trigonometry as follows: 

ZPfinal-ZY 

sin. 
= (3.27) 

L 
. 
= sin-1 (-cos( )sin()) (3.28) 

3.5.4 Comparing the equations – position equations 
In each of the rotation sets, the x-coordinate is identical. The sets differ in the y-
and z-coordinate sets as follows: 

YZ y-coord: L 
cos()sin( ) from Equation 3.24 
ZY y-coord: L 
sin( ) from Equation 3.26 

The y-coordinates differ by a factor of cos()between the YZ and ZY systems. 

YZ z-coord: -L 
sin() from Equation 3.24 
ZY z-coord: -L 
cos( )sin() from Equation 3.26 

The z-coordinates differ by a factor of cos( )between the YZ and ZY systems. 

The two sets of 3D rotations are different because of the commutative rules of 
matrix multiplication, in general when multiplying two rotation matrices A and B 
the matrix product AB will be different from BA. 

3.5.5 Comparing the equations – kinetic energies 
Comparing the differences in the kinetic energies of the spinning systems in a local 
coordinate system: 

m

YZ Ek 
= 2 L2 .2 + cos()2  .
2 

m 
L2 .2 2

ZY Ek 
= 2  2 + cos( ).
Clearly, the two are similar – the difference essentially being the substitution of 
one angle for the other between the two sets of equations. 
59 


3.5.6 Comparing the equations – potential energies 
Comparing the differences in the potential energies of the spinning systems in an 
inertial coordinatesystem isastagemorecomplicated,asthe local coordinatesystem 
mustbetranslatedfromthe local centretothe inertial centreto incorporatethefull 
inertial terms. 

The potential energy term for each mass point j 


µmj

Ep 
= v 

pj 
· pj 


v 

depends strongly on the magnitude of distance, given by pj 
· pj. This evaluates to 
the following for YZ and ZY rotations10 

YZ: pj 
= {R 
+ L 
cos()cos( ),L 
cos()sin( ), 
-L 
sin()} 

ZY: pj 
= {R 
+ L 
cos()cos( ),L 
sin( ), 
-L 
cos( )sin()} 

Therefore 

YZ: v pj 
· pj 
= L2 + R2 +2LR 
cos()cos( ) 

ZY: v pj 
· pj 
= L2 + R2 +2LR 
cos()cos( ) 

For the potential energy term: 

YZ Ep 
= v µmj 


L2+R2+2LR 
cos()cos( ) 

ZY Ep 
= v µmj 


L2+R2+2LR 
cos()cos( ) 

That is, the potential energy expression of the two sets of equations YZ and ZY 
are identical for an inertially planar system. 

When the orbital rotation . 
is introduced to the potential energy equations, the 
result will be identical to the expression above, as the inertial rotation through R,Z 
is a rotation about one axes only, unlike thedual system of rotations outlined above. 

Theexpressionforpotential energy willchangesignificantly whenrotated through 

10With a simple offset from the origin by R, representing the orbital radius. 

60 


around the orbit in 3D when the orbital parameter . 
is included. 

3.5.7 YZ rotation in inertial axes 
The full rotation sequence in inertial axes, for a rotation of an initial vector through 
a Y-axis rotation, then a Z-axis rotation, translations in CoM and radius vector, 
then a final rotation about the Z-axis is as follows: 

Pfinal = R,Z · (P0.CoM -Pfacility.CoM +(R ,Z · R,Y · P1)) (3.29) 

3.5.8 Singularities in the rotational systems 
The main disadvantage in using the method of rotational matrices is they have 
singularities or ‘gimbal lock’ points where there is a lack of a unique output when a 
rotation is performed that aligns it with the rotational axis. 

In the case of YZ rotation, an initial rotation of 90. about the Y-axis would align 
the original vector with the Z-axis. A second rotation about the Z-axis would be 
redundant as the vector would not change value. 

In the case of ZY rotation, an initial rotation of 90. about the Z-axis would align 
the original vector with the Y-axis. A second rotation about the Y-axis would be 
redundant as the vector would not change value. 

In general, any rotation through 90., 270. (or any multiples thereof) so that the 
original vector aligns itself with the new axes of rotation will produce a singularity. 

The equations of motion usually can not be numerically integrated with the initial 
condition as a singularity; the lack of uniqueness of the solution violates the method 
of using the Lagrangian to solve the equations of motion. 

61 



3.5.9 Practical uses of the different equations 
The two sets of equations above may find different applications, depending on the 
desired investigative outcome required. 

If a system of equations is needed to solve a solution space near an axis, then 
naturally, the system of equations should be chosen to avoid a singularity. As the 
choice of system of equations is entirely arbitrary, the problem of singularities has 
been avoided here. 

Quaternions were considered as an alternative to the three-angle rotation system 
above -they confer several advantages such as the ability to multiply the rotations 
inonepolynomial and they arenotpronetogimbal lock. However,oneof themain 
disadvantages of quaternions is that it is very difficult to extract physical meaning 
fromthe individual componentswithout asignificantdegreeofpost-processing. For 
this reason, it was decided to remain with the three-angle rotation method. 

The YZ system of equations is slightly more compact and easier to interpret than 
the ZY equations, although both give identical results when integrated. The difference 
in time taken to solve the two sets of equations differs by approximately 5% 
using a wide range of initial conditions. 

3.6 Numerical integration techniques 
The equations of motion, once assembled, are integrated with Mathematica11 using 
the NDSolve subroutine. This is a Jack-of-all-trades solution that effectively deals 
withtheproblems of choosingtimesteps, any stiffness that may arise and a multitude 
of otherproblemsthatplaguetheresearcher. However, caremustbetakentoassure 
that the output of the numerical integrator is representative of real life – validation 
and sanity-checking at allstageswillnot ensurepoorresultsareentirely eliminated, 

11Specifically, using Mathematica version 5.1 

62 


but they will reduce the risk of spurious results. 

The generated equations are solved with Mathematica’s general purpose solver 
NDSolve[. . .] with the following parameters: 

. 
AccuracyGoal -> 
Automatic, PrecisionGoal -> 
Automatic – controls 
the number of digits of accuracy, effectively the absolute and relative errors 
respectively. 
. 
With PrecisionGoal->p and AccuracyGoal->a, Mathematica attempts 
to make the numerical error in a result of size x 
be less than 10-a 
+ |x|10-p. 

. 
WorkingPrecision -> 
20 – limits internal computations to 20-digit precision. 
From the online Mathematica NDSolve FAQ webpage – [Wolfram, 2007]: 

“For initial valueproblems,NDSolve uses anAdamsPredictor-Corrector 
method for non-stiff differential equations and backward difference formulas(
Gearmethod)forstiffdifferential equations. It switchesbetween 
the two methods using heuristics based on the adaptively selected step 
size. It starts with the non-stiff method under essentially all conditions 
and checksfor the advisability of switching methods every10 or20 steps. 

The algorithms and the heuristics for switching between algorithms 
are described in the following references: [Hindmarsh, 1983], [Petzold, 
1983]” 

More in-depth information about the NDsolve functions is also available from the 
Mathematica Journal: [Keiper, 1992]. 

63 



Chapter 4 

Dynamics of tethers with 
inclination 

The simple dumbbell tether model can be expanded to include an inclination 
component to the orbit. 

Theprocessofaddingthe inclinationtermwillbeexemplified with atetherpayload 
deliverytotheMoonunderatime-varyinginclinationgivenbytheSaroscycle,which 
describes the inclination variation of the Moon’s orbit around the Earth. 

The Saros cycle describes the Moon’s orbital inclination cycle, lasting 18 years 11 
daysand8hours,cutting theSun’seclipticplane(usedasthe inertial coordinate 
plane). This cycle determines, in the short term at least, the Lunar and Earth’s 
eclipse frequencies and locations. This also allows the inclination of the Moon to 
be accurately predicted from ephemeral data. Figure 4.1 shows the inclination as a 
function oftime over oneSaros cycle withdata takenfrom theJPLHorizons website 
[JPL, 2008]. 

As shown in Figure 4.1, the Saros cycle varies from a maximum inclination with 
respect to thegeocentricplane of approximately 28.8. to a minimum of18.0., which 

64 



Moon DEC_(ICRF/J2000.0) 

-30 
-20 
-10 
0 
10 
20 
30 
Moon Declination (deg) 
01/01/2024 
01/01/202301/01/202201/01/202101/01/202001/01/201901/01/201801/01/201701/01/201601/01/201501/01/201401/01/201301/01/201201/01/201101/01/201001/01/200901/01/200801/01/200701/01/200601/01/200501/01/2004

Date 

Figure 4.1: Ephemeris of Lunar orbital inclination [JPL, 2008] 

poses a problem to the trajectory and mission specialists of a long-term reusable 
launch system, like the MMET. The position of the Moon changes, which makes 
it more difficult to launch a Lunar mission, and the cost of the mission will change 
due to a potentially expensive1 plane change manoeuvre. 

Comparing the Lunar inclination to Earth’s tilt of 23.5., the Moon’s inclination 
with respect to Earth’s equatorial plane varies by approximately ±5. . The 
V 
required to perform an inclination change will be small, but the dynamics of the 
system will depend on the inclination with respect to the inertial plane. In 2008, 
this is approximately 28. . 

4.1 Addition of inclination to tether model 
The expansion of the tether model to include an inclination term is relatively 
straightforward, at least in terms of the model outlined in Section 3.5.7. The inclination 
term is added with a rotation matrix2 R,Y before the final inertial rotation 

1In terms of fuel and therefore cost. 
2Rotating around the Y-axis through the inclination angle, . 


65 


sequence R,Z is performed. 

Thefullrotationsequence in inertialaxes,forarotationof an initial vectorthrough 
a local Y-axis rotation, then a local Z-axis rotation, translations with the CoM and 
radiusvectors,thentworotationsaboutthe(recentlycreated) inertialY-axis,then 
the inertial Z-axis is as follows: 

Pfinal = R,Z · R,Y · (P0.CoM -Pfacility.CoM +(R ,Z · R,Y · P1)) (4.1) 

As inSection3.3,Lagrange’sequationswillnowbederivedforthesystem including 
the orbital inclination. 

4.1.1 5 degree of freedom model 
Five variables of interest are chosen – {R, 
, 
, 
 , 
}– thatfulfil all the requirements 
of generalised coordinates. An analysis will be performed to investigate the effect 
the inclination variable has on the tether system orbit. 

The mathematical model will be constructed to include a symmetrical dumbbell 
tether with a facility mass at the centre of the system. The important mass points 
included in the model are the two identical payload masses, Mpayload, the facility 
mass, Mfacility, and the tether masses, Mtether, which are concentrated at the centre 
of each tether. 

The orbital system {R, 
, 
}is conceptually separated from the local rotation system 
{ , 
}. The equations of motion for the orbit are constructed as one concentrated 
point mass, and the local rotational system is superimposed on this orbit. 
This allows a significantly simpler representation of the equations, however, this 
does not allow the orbital motion to be influenced by the local rotations. Overall, 
this assumption is valid and should give an insight into the broad motions of the 
MMET on an inclined orbit. 

66 


The stator is not included in this model because this would significantly increase 
the complexity of the system. 

4.2 Constructing the equations of motion 
The sequence of rotations follows the YZ system of rotations as described earlier in 
Section 3.5.2. 

Manipulating the position vector from the original Equation 4.1, the rotation 
matrices are collected through the commutative properties of matrices and terms 
containing the Radius vector, P0.CoM, and the local vector3 , P1, are separated: 

Pfinal = R,Z · R,Y · (P0.CoM -Pfacility.CoM +(R ,Z · R,Y · P1)) (4.2) 

Pfinal = R,Z · R,Y · (P0.CoM -Pfacility.CoM)+R,Z · R,Y · R ,Z · R,Y · P1 (4.3) 

Taking the assumption that the dumbbell is symmetric about the facility mass: 

. 
. 
. 
. 
CoMX 0 
. 
. 
. 
. 
. 
. 
. 
. 


. 
...

Pfacility.CoM =0 . CoMY =0 

. 
... 


. 
... 


CoMZ 0 

gives the simplified positional equation Pfinal for a single mass point, P1 in the 
inertial coordinate system: 

Pfinal = R,Z · R,Y · (P0.CoM)+R,Z · R,Y · R ,Z · R,Y · P1 (4.4) 

3The local vector is the position vector between the facility and payload 1. 

67 



When evaluated, this becomes: 

... 
. 


cos . 
cos. 
cos(cos. 
cos . 
cos. 
-sin. 
sin )-cos. 
sina 
sin. 


... 
. 


... 
. 


... 
.

Pfinal = R 
cos. 
sin . 
+ L 
cos(cos. 
cos. 
sin. 
+ cos . 
sin  )-sina 
sin . 
sin . 


. 
. 
-sin. 
. 
. 
. 
. 
-cos . 
sina 
-cosa 
cos. 
sin. 
. 
. 
(4.5) 

The purpose of separating the inertial and local systems in Equation 4.5 is that 
Lagrange’s equations are easier to derive and subsequently, the equations of motion 
are computationally more efficient to solve. 

4.2.1 Rotations in the local coordinate system 
Figures 4.2a, 4.2b, 4.3a and 4.3b show the sequence of rotations needed to assemble 
Equation 4.4: R,Z · R,Y · R ,Z · R,Y · P1. 

Figures 4.2a and 4.2b show the steps needed to assemble the local coordinate 
system, with Figure 4.2a duplicating the steps followed to arrive at the YZ rotation 
sequence followed in Section 3.5.2. This is followed by a translation through 
P0.CoM = {R, 
0, 
0}to transfer the local coordinate system to the inertial coordinate 
system. Figures 4.3a and 4.3b show the tether rotating around the inertial axes 
system – first through a rotation of . 
around the inertial Y-axis, then a rotation of 
. 
around the inertial Z-axis. 

Figure Rotational sequence 

Figure 4.2a R ,Z · R,Y · P1 
Figure 4.2b R+ R ,Z · R,Y · P1 
Figure 4.3a R,Y · (R+ R ,Z · R,Y · P1) 
Figure 4.3b R,Z · R,Y · (R+ R ,Z · R,Y · P1) 

Table 4.1: Reference table with figure and corresponding rotation 

68 



Z 

y 
a 
Y 
y 
a 
a 
ZYZYZXZXYZXYZY(a)YZ local rotation – R,Ythen R ,Z
X 

y 
y 
q 
a 
i 
i 
a 
a 
Y 
X 
Y 
Z 
Z 
X 
XYZYZZYZXZXYZY
(b)Translation through R along X-axis 

Figure 4.2: Rotation sequence for YZ rotation with inclination. a 
is shown as a 
negative rotation here for ease of visualisation. 

69 



y 
Z 
Y 
X 
i 
a
y 
a 
y 
y 
q 
a 
i 
i 
a 
a 
Y 
X 
Y 
Z 
Z 
X 
ZYXYZYZYZXZXYZXYZYXYZYZZYZXZXYZY
y 
Z 
Y 
X 
i 
a
y 
a 
y 
y 
q 
a 
i 
i 
a 
a 
Y 
X 
Y 
Z 
Z 
X 
ZYXYZYZYZXZXYZXYZYXYZYZZYZXZXYZY
(a)Y inertial rotation through angle . 
– R,Y 


X 
Y 
Z 
q 
a 
a
y 
ay 
y 
Z 
Y 
X 
i 
a
y 
a 
y 
y 
q 
a 
i 
i 
a 
a 
Y 
X 
Y 
Z 
Z 
X 
XYZZYZYZXZXYYZZYZYZXXZZYXYZYZYZXZXYZXYZYXYZYZZYZXZXYZY
(b)Z inertial rotation through angle . 
– R,Z 


Figure 4.3: Rotation sequence for YZ rotation with inclination. a 
and . 
are shown 
as negative rotations here for ease of visualisation. 

70 



4.2.2 Lagrange’s equations 
Oncetheposition of the masspointshavebeendefined,theprocess of constructing 
the equations of motion will follow that outlined in Section 3.3. 

As the equations form two separate groups, this will be examined separately: in 
Section 4.2.3 for the local system and Section 4.2.4 for the orbital system. The 
equations are combined and listed in Appendix A. 

4.2.3 Lagrange’s equations – local system 
The mass points, as indicated earlier, are: 

. 
the two payload masses, each a distance L 
from the facility mass and each 
with mass Mpayload 

. 
the two tethers with masspoints each adistance L/2,halfway alongthetether, 
each with mass Mtether 

. 
the facility mass, Mfacility. 

Thetethersystem issymmetrical;thein-planeangleswillbedefined as . 
and - , 
while the out-of-plane angles will be defined as a 
and -a 
respectively. 

Aspreviously,thetethersareassumed toberigidand inextensibletosimplify the 
equations. The stator will not be included to simplify the equations. The masses of 
the payloads are assumed to be equal and the masses of the tethers are assumed to 
be equal such that: 

Mpayload 1 = Mpayload 2 = Mpayload and Mtether 1 = Mtether 2 = Mtether 

71 



The locations of each of the mass points in the inertial plane are therefore: 

... 
. 


cos. 
cos. 
cos(cos. 
cos. 
cos. 
-sin. 
sin )-cos. 
sina 
sin. 


... 
. 


... 
. 


... 
.

Ppayload 
1 
= R 
cos. 
sin. 
+ L 
cos(cos. 
cos. 
sin. 
+ cos. 
sin )-sina 
sin. 
sin. 
(4.6) 

... 
. 


... 
. 


-sin. 
-cos. 
sina 
-cosa 
cos. 
sin. 


... 
. 


cos. 
cos. 
cos(cos. 
cos. 
cos. 
-sin. 
sin )-cos. 
sina 
sin. 


... 
. 


... 
. 


... 
.

Ppayload 
2 
= R 
cos. 
sin. 
-L 
cos(cos. 
cos. 
sin. 
+ cos. 
sin )-sina 
sin. 
sin. 
(4.7) 

... 
. 


... 
. 


-sin. 
-cos. 
sina 
-cosa 
cos. 
sin. 


... 
. 


cos. 
cos. 
cos(cos. 
cos. 
cos. 
-sin. 
sin )-cos. 
sina 
sin. 


... 
. 


.. 
L 
.. 


... 
.

Ptether 
1 
= R 
cos. 
sin. 
+ cos(cos. 
cos. 
sin. 
+ cos. 
sin )-sina 
sin. 
sin. 
(4.8) 

... 
.

2 

... 
. 


-sin. 
-cos. 
sina 
-cosa 
cos. 
sin. 


... 
. 


cos. 
cos. 
cos(cos. 
cos. 
cos. 
-sin. 
sin )-cos. 
sina 
sin. 


... 
. 


.. 
L 
.. 


... 
.

Ptether 
2 
= R 
cos. 
sin. 
- cos(cos. 
cos. 
sin. 
+ cos. 
sin )-sina 
sin. 
sin. 
(4.9) 

... 
.

2 

... 
. 


-sin. 
-cos. 
sina 
-cosa 
cos. 
sin. 


.. 


cos. 
cos. 


.. 


.. 


..

Pfacility 
= R 
cos. 
sin. 
(4.10) 

.. 


.. 


-sin. 


The velocities of each of the mass points are then calculated. These are not listed 

here because of the length of the equations. 

The fivemasspointsareused to find thetranslationalkineticenergy, Ttrans, of the 

system: 

Mpayload 1 Mpayload 2 
Ttrans = Vpayload 1 · Vpayload 1 + Vpayload 2 · Vpayload 2 


22 

Mtether 1 Mtether 2 

+ Vtether 1 · Vtether 1 + Vtether 2 · Vtether 2 
22 
1 


+(Mpayload 1 + Mpayload 2 + Mtether 1 + Mtether 2 + Mfacility)VCoM · VCoM (4.11) 
2 

72 



which evaluates to: 

1 

Ttrans = cos 2().2 +.2 R2 + R.
2 (MFacility +2(Mpayload + Mtether))+ 

2 

1 
... 


L2 16.2 +32.
sin()sin( ).a 
-4 -2cos(2 )cos2()+cos(2)-3 .2+ 

64 

16cos2() .
2 +16.
 .
2cos2()cos()-cos( )sin(2)sin()+ 

.2 -8cos2()cos(2 )sin2()+2cos(2)-2cos(2)+ 
3cos(2(. 
-))+3cos(2(a 
+ ))-8cos( )sin(2)sin(2)+10)+ 


. 
. 
. 


16..sin()sin(2 )cos2()+2cos( ).a 
+ cos().+  .
sin(2)sin( ) 
(4Mpayload + Mtether) (4.12) 

The rotational kinetic energy, Trot, of the system is found in a similar way: 

Trot =(Ipayload 1 + Ipayload 2 + Itether 1 + Itether 2)· (!local · !local) (4.13) 

where: 

.. 
.. 


0 100 

.. 
.. 


.. 
.. 


.. 
..

!local = .I3 = 010 

.. 
.. 


.. 
.. 


.

. 
001 

Ipayload 1 =L2Mpayload 1 · I3 Ipayload 2 =L2Mpayload 2 · I3 
Itether 1 =
1 
L2Mtether 1 · I3 Itether 2 =
1 
L2Mtether 2 · I3


33 

The evaluated sum of the component rotational kinetic energies are: 

1 

Trot = L2 (3Mpayload + Mtether) .2 +  .
2 (4.14) 

3 

73 



The sum of the potential energies, U, are: 

nn 


µmj 
µmj

U 
= 
. 
= 
. 
(4.15) 

Pj 
Pj 
· Pj
j=1 j=1 

where Pj 
are the magnitude of the mass point locations, given in Equations 4.6 to 

4.10. 
The full rotational sequence is not needed here, merely the distance between the 
two points, which leads to a simpler expression of the magnitude of the distance. 
Taking Equation4.2asthestartpoint,thedistance(i.e.,themagnitude) is: 

. 
Ppayload 1 
. 
= 
. 
R,Z · R,Y · (P0.CoM + R ,Z · R,Y · P1)
. 
(4.16) 

. 
Ppayload 1 
. 
= 
. 
R,Z
. 
· . 
R,Y
. 
· . 
P0.CoM + R ,Z · R,Y · P1
. 
(4.17) 

The magnitude of a purely rotational matrix is, by definition, equal to one: 

. 
R,Z
. 
=1 (4.18) 

. 
R,Y
. 
=1 (4.19) 

This simplifies Equation 4.17 to: 

. 
Ppayload 1 
. 
= 
. 
P0.CoM + R ,Z · R,Y · P1
. 
(4.20) 

... 
. 


. 
R 


L 
cosa 
cos. 


... 
.. 


... 
.. 


... 
.. 


. 
Ppayload 1 
. 
=0+ L 
cosa 
sin. 
(4.21) 

... 
.. 


... 
. 


. 
0 -L 
sin a 
. 


Ppayload 1 = L2 +2LR 
cosa 
cos. 
+ R2 (4.22) 

74 



The evaluated expression for the potential energy is therefore: 

U 
= - . 
µ 
Mfacility . 
-µ 
. 
v 
1 
+ v 
1 . 
Mpayload 

R 


L2 -Lmed + R2 L2 + Lmed + R2 

11 

-2µ 
v + v Mtether 
L2 -2Lmed +4R2 L2 +2Lmed +4R2 

(4.23) 
where: 

Lmed =2LR 
cosa 
cos. 


The Lagrangian is then given by L 
= T 
-U: 

2  2

L2 (3Mpayload + Mtether) .+.
L 
= Ttrans + 

3 

µMfacility 11 

++ µ 
v + v Mpayload 

R 


L2 -Lmed + R2 L2 + Lmed + R2 

11 

+2µ 
v + v Mtether (4.24) 
L2 -2Lmed +4R2 L2 +2Lmed +4R2 
where Ttrans is given by Equation 4.12. 

Lagrange’s equations are then generated for two out of the five generalised co


ordinates, { , 
}as follows: 

d 
@T 
@T 
@U 


- += Qj 
(4.25) 

dt 
@q.j 
@qj 
@qj 


and are provided in Appendix A along with the non-conservative forcing terms. 

4.2.4 Lagrange’s equations – global system 
The global equations used to construct the motion of the facility around the Earth 

are now outlined. The mass for the system is summed and used as the total mass 

75 



located at the CoM: 

Mtotal =2Mpayload +2Mtether + Mfacility (4.26) 

The locationsof thesinglerepresentativemasspoint inthe inertialplane istherefore: 


.. 


R 
cos. 
cos . 


.. 


.. 


..

Porbit = R 
cos. 
sin. 
(4.27) 

.. 


.. 


-R 
sin . 


The velocities ofeach of the masspoints are then calculated4 . Equation4.28 shows 
the velocity of the system CoM: 

.. 


.

R 
cos. 
cos. 
-R.
cos. 
sin. 
+.. 
cos. 
sin. 


.. 


.. 


..

Vorbit = R.cos. 
cos. 
+ sinR.
cos. 
-R.sin. 
(4.28) 

.. 


.. 


-R.cos. 
-R.
sin . 


Thesinglemasspoint isused to find thetranslationalkineticenergy, Ttrans, of the 
system: 

Mtotal 

Ttrans =(Vorbit · Vorbit) (4.29) 

2 

Mtotal 

Ttrans = .2 cos 2()+.2 R2 + R.
2 (4.30) 

2 

The rotational kinetic energy is assumed to be zero: 

Trot =0 (4.31) 

4Note: as the inclination term is given the symbol iota, , the first derivative of the inclination 
term, , is., and not the ninth letter of the Latin alphabet, i. 

76 



The expression for potential energy, U, is simply: 

µMtotal µMtotal 

U 
= = (4.32) 

|R.R| R 


as the distance to the CoM is defined as R. 
The Lagrangian is then given by L 
= T 
-U: 

Mtotal µMtotal 

L 
= .2 cos 2()+.2 R2 + R.
2 - (4.33) 

2 R 


Lagrange’s equations are then generated for three out of the the five generalised 

coordinates, {R, 
, 
}as follows: 
. 
. 
d 
@T 
@T 
@U 
dt 
@q.j 
- @qj 
+ 
@qj 
= Qj 
(4.34) 

and are provided in Appendix A along with the non-conservative forcing terms. 

4.2.5 Non-conservative forces 
TheMMET system requires that aforcebe transferredfrom the motor to the tether 
arms to ‘spin-up’ the system. The mechanics of spin-up of such a system is nontrivial, 
especially when flexural5 terms and inertial terms are taken into account. 
The spin-up dynamics of length deployment is covered in Chapter 5. 

Thetetherarmsareassumed toberigidand symmetrical at alltimes, limiting the 
payload tethers to a planar disc. The motor torque is assumed to act in the same 
plane as this disc. The stator arms are neglected. 

The motor torque will exert a large power drain, therefore all the power for the 
motor willbederivedfrom the solar arraysharnessing afree,plentiful and renewable 
source of energy. The Forcing term in Lagrange’s equations must reflect the binary 

5Flexural terms are not included in the analysis due to the complexities they introduce. 

77 


nature of the availability of the power supply. The Eclipse function modulates the 
motor torque to take into account the eclipsing of the Sun by the Earth, reducing 
the torque to t 
= 0 when in darkness and taking the default t 
= max when in 
sunlight. 

Eclipse is the binary function: 

cos. 
cos> 
0 

Eclipse = 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
1 
. 
. 
. 
. 
. 
. 
. 
R 
cos . 
2 
.R2 = R2 
. 
Earth 
. 
. 
RSun 
. 
. 
. 
. 
. 
. 
. 
. 
1-cos2()cos2() 1+ 
AND 
(4.35) 

. 


. 


. 


. 


. 


. 


. 


. 


. 


. 


. 


. 
0 otherwise 

The radius of the Earth’s shadow is scaled from REarth by the factor RSun+R 
cos . 


RSun 

as shown in Figure 4.4. The hypotenuse of the Y-and Z-component of the CoM at 
a time, t, in the orbit path is calculated and checked. If it is less than the shadow 
distance, then the CoM is in shadow. 

RSun + R 
cos. 


RShadow = REarth 

RSun 

. 
(4.36) 

R 
cos. 


RShadow = REarth 1+ 

RSun 

Y 
2 22 

=(R 
cos . 
sin)+(R 
sin )

CoM inertial + Z2 

CoM inertial 

(4.37) 
= R2 (1-cos2()cos2()) 

Hence, the eclipse checking function is calculated as: 

2 

... 
R 
cos()

R2 1-cos2()cos2() = R2 1+ (4.38) 

Earth 

RSun 

which gives two eclipses an orbit, therefore an additional constraint is added to 

78 



Y 


Z 

Sun 

CoM Orbital Path 

Cone of Eclipse 
EarthR 
Earth 
P 
X......................
....














..................









..............i q..............
























..
....





RSun 

Figure4.4:Eclipse checkingfunction(notto scale) 

ensure that the CoM is eclipsed only once per orbit: 

cos. 
cos > 
0 (4.39) 

A diagram of the distances on-orbit is shown in Figure 4.4 and the Mathematica 
output for a 29. orbit is shown in Figure 4.5. The Sun is assumed to be a point 
source and the Earth is assumed to describe a circular orbit around the sun. 

Eclipse 
EclipseEclipse1 
11

0 
00


Orbits 
OrbitsOrbits

1
113 
33

0
001
112 
22

..
.....
... 
....
.....
... 
..

2
222 
22

Figure 4.5: Sample plot of the Eclipse function in Mathematica 

79 



4.3 Analysis of tether on an inclined orbit 
The generated equations are solved with Mathematica’s general purpose solver 
NDSolve[. . .] with the following parameters: 

AccuracyGoal -> 
Automatic 
PrecisionGoal -> 
Automatic 
WorkingPrecision -> 
20 


To fully understand the interaction between the local and inertial systems, Lagrange’s 
equations are solved with different initial conditions or ‘cases’. These are 
given in Table 4.2. 

.

case .  .. 
a 
eccent t 


I1 0 0 0 0 0 0 0 1000 
I2 0 0 0 29/180 0 0 0 1000 
I3 0 1 0 29/180 0 0 0 1000 
I4 0 0 0 29/180 0 -29/180 0 1000 
I5 0 1 0 29/180 0 -29/180 0 1000 

Table 4.2: Initial conditions for the inclination numerical solutions. 

Case I1 gives a baseline in the inertial plane: no inclination, no out-of-plane spin 
and no initial in-plane spin. Case I2 introduces inclination, while case I3 has an 
inclined orbitwith asignificant initial in-planespin. CasesI4 andI5 arecomparable 
tocasesI2 andI3,with thedifferenceof anout-of-planespinequal tothe inclination 

(i.e. so the spin-plane is coincidentalwith the inertial plane). 
All five cases have an equal torque level to enable a fair comparison. 

The initial conditions6 for the other parameters held constant are: 

L 
=1000m, 
Mpayload =10kg, 
Mtether =100kg, 
Mfacility =500kg, 
e 
=0, 


R.
[0] =0,.[0] =0.00113905rad/s, 
[0] =0,R[0] =7.378* 106 m 


6Note: the inclination term is represented by the Greek letter iota – . 
– therefore the first 
derivative with respect to time is iota dot – . – and the second derivative iota dot dot – ¨. This 
.

was defined in order to remove the confusion between i dot and iota dot – i, 
.. 

80 


Figures4.11 to4.18 show the solution to the equations of motionfor10 orbits over 
63069.5 seconds where: 

. 
Figure 4.11 shows plots of a 
vs. time 

. 
Figure 4.12 shows plots of .a 
vs. time 

. 
Figure 4.13 shows plots of a 
vs. . 


. 
Figure 4.14 shows 3D phase plots of . 


. 
Figure 4.15 shows 3D phase plots of a 


. 
Figure 4.16 shows plots of 
V 
vs. time 

. 
Figure 4.17 shows plots of tension vs. time 

. 
Figure 4.18 shows plots of stress vs. time 

4.3.0.1 Behaviour of R, . 
and . 
with time 
The orbital parameters R, . 
and . 
are shown in Figure 4.10. They are deliberately 
chosensothattheorbitalparametersareindependentfromthe localparameters,as 
such the MMET does not influence any of these generalised coordinates. 

Together, the three generalised coordinates R, . 
and . 
describe a circular orbit in 
the inertial plane in case I1 and a circular inclined orbit in cases I2 to I5. 

4.3.0.2 Behaviour of a 
with time 
Figures 4.11a to 4.11e demonstrate the a 
vs. time behaviour of the system. Clearly, 
the baseline case I1 shows there is no interaction between the inclination and out-
of-plane coordinates when both are initially zero. 

Comparing CasesI2 andI3 inFigures4.11b and4.11cgivesan ideaof therelative 
.

stabilising influence of the in-plane spin velocity,  . It is the gyroscopic action of 
the in-plane spin that limits the out-of-plane coordinate, . 

The opposite appears to be true when Figures 4.11d and 4.11e – cases I4 and I5 

81 



– are compared. The out-of-plane spin angle is set such that the local system is 
aligned with the inertialplane. Whenthis isacceleratedfromzerovelocity initially, 
the out-of-plane angle returns to be coplanar with the inclination orbit7, with a 
small nutation of(0.1 rad 
˜ 6.). When started from a large initial in-plane velocity 
in the inertial plane, the opposite is true – it remains spinning in the inertial plane 
with a small adjustment towards the orbital plane. 
4.3.0.3 Behaviour of .with time 
Figures 4.12a to 4.12e demonstrate the .vs. time behaviour of the system. As 
previously, the baseline case I1shows there is no interaction between the inclination 
and out-of-plane coordinates when both are initially zero. 

Case I2 and I4 show the angular rate increasing from zero – this is larger than 
desired for operating the MMET, especially in the early start-up regime. However, 
case I3 shows that the greater the in-plane spin velocity, the lower the out-of-plane 
nutation. 

4.3.0.4 Behaviour of a 
with . 
Figures 4.13a to 4.13e show the .a 
vs. . 
interaction of the system. 

The graphs here are inconclusive. There is a large interplay between a 
and . 
.

in cases I3 and I5, where the in-plane velocity . 
is large, and a correspondingly 
.

smaller relationship between the two in cases I2 and I4 where the . 
term is zero. 
This, however, does not point to any concrete notions of interdependence. 

CaseI4 clearly showstheprogression of a 
froma largeout-of-planemotion(starting 
in the bottom right of the graph) to a smaller angle as the in-plane velocity 
increases. 

7The inclination orbit is the orbital plane, offset 29/180rad 
from the inertial plane. 

82 


.¨ 

4.3.0.5 Behaviour of  , 
 , 
. 
The eclipse functionality is clearly shown in the . 
phase plots in Figires 4.14a to 
4.14e and the plot of 
V 
in Figure 4.16a to 4.16e. 

The . 
phase plane plots illustrate the shadow periods of the graph where the in-
plane angular acceleration  ¨, shown in the horizontal axis, is approximately zero 
compared to the illuminated periods where the tether is accelerated by the motor. 

Theplots of
V 
highlight the end result of the work done by the motor – the 
V 
thesystemcan imparttothepayload. Inthiscase,a
V 
of approximately 400m/s 
is attained for a modest acceleration period of 17.5 hours. 

In cases I1 and I3, the out-of-plane coordinate a 
is zero and the rate of increase 
.

in . 
is highest. The greater the out-of-plane spin angle, the less energy is available 
totransfer into increasing the(useful) in-planespinrate. 

Cases I4 and I5 show the more disordered plots when the out-of-plane coordinate 
is non-zero. Clearly, having an clean acceleration profile is advantageous from a 
stress-minimisation point of view – these profiles have a spin-up phase that feature 
a large .a 
term, which is difficult to aim when the payload must be released. 

Similarly, case I2 does not have a clean spin-up phase when compared to case I1. 
This ispurelyduetothe largeout-of-planemotionthat interfereswith the in-plane 
spin. 

4.3.0.6 Behaviour of , 
, 
.¨
The a 
phase plane behaviour of the system is shown in Figures 4.15a to 4.15e. 
In case I3 – high  .
and the out-of-plane initial angle, , set to zero – the out of 
plane velocity is minimised. Marginally larger out-of-plane angular velocities and 
accelerations are shown by case I4: zero initial  .
and the out-of-plane initial angle 
set to coincide with the inertial plane. 

83 


4.3.0.7 Behaviour of 
V 
with time 
The 
V 
vs. time behaviour of the system is shown in Figures 4.16a to 4.16e. 

The calculation of
V 
isstraightforward, it istheproduct of thetether length and 
the angular velocity. The maximum useful 
V 
is obtained when the out-of-plane 
angle is zero, and is analysed in Section 4.4. In this case, the maximum useful 
V 
is simply the product of length and angular speed: 

.


V 
= L. 
(4.40) 

As thegraphs of 
V 
show the magnitude of the resultant velocity, cases I1 I2 and 
I4 should be almost identical, and cases I3 and I5 should also be identical. Whereas 
the former is true, case I5 has a useful velocity that is lower than expected. This 
is caused by rapid changes in the out-of-plane velocity ., which detracts from the 

.

in-plane spin velocity,  . 

4.3.0.8 Behaviour of tension with time 
The tension profile of the tether is shown in Figures 4.17a to 4.17e. 

Thecaseswith zeroinitialin-planespin – casesI1,I2 andI4 –exhibitsimilarlevels 
of tension in the tether, approximately 10 kN. The tension appears to be slightly 
lower in the baseline case I1, however, this may be due to the larger out-of-plane 
movements in cases I2 and I4. 

The remaining cases I3 and I5 start with a higher tension due to their large initial 
in-plane spin rate. Similarly to the 
V 
plot in Figure 4.16e, the tension in case I5 
rapidly fluctuates due to the large out-of-plane angular motions. 

4.3.0.9 Behaviour of stress with time 
The stress profile of the tether is shown in Figures 4.18a to 4.18e. 

84 


Thegraphsherearevery similartothetensiongraphs,asstress isequal totension 
divided by the cross-sectional area, 0.000064 m2 (equal to an 8mm 
square section 
of tether). 

4.3.1 Analysis of tension and stress in tether 
The tension in the tether ignores any out-of-plane motion of the tether, as it was 
shown that the out-of-plane motion is not significant for small deviations from the 
spin-plane. 

The tension in the tether is calculated as: 

Lpayload 

. 2

Tpayload = Mpayload  2Lpayload + A 
.(4.41) 

2 

with the firsttermasthetensionduetothepayload,thesecond termasthetension 
duetomassof thetether itself. Thesecond termdominates inthiscase,asthemass 
of the tether is an order of magnitude higher than the mass of the payload. 

The stress in the tether is calculated as: 

Tpayload 

s 
= (4.42) 

A 


Themaximumallowablestress inthetether iscalculatedusingtheUltimateTensile 
Stress(forZylon,theUTS is5.9* 109 N/m2)and aSafety Factor(SF=1.5) as: 

s 
5.9* 109 

max 
== =3.933GP 
a 
(4.43) 

SF 1.5 

The variables are defined as: 

Mpayload =10kg, L 
=1000m, . 
=1570kg/m3 , A 
=0.000064m2 

using values ofthetether8 composed ofZylon. 
fibres withpropertiesfromTableD.1, 

8The area is equivalent to a rectangular extrusion of 0.008m. 

85 


Appendix D. 

Themaximumallowablestressgivesanabsoluteceilingtothestresses inthetether. 
This maximum stress, working backwards through the tension, is dependant on the 
in-plane velocity assuming the material and geometric properties of the system do 
not change when it is in-orbit. This interplay between  .
and Lpayload 
drives the 
maximum 
V 
and therefore the entire feasibility of the system. 

4.4 Tether release aiming 
The criteria to maximise the 
V 
when using the motorised tether with a Hohmann 
two-burn trajectory are as follows. 

The tether will gain the most velocity from the release if the velocity vector is at 
right-angles to the orbital plane9 . This is generally satisfied when the tether arm is 
aligned with the radius vector, a criteria of . 
. 
0 and a 
. 
0 

An excellent treatment of the sensitivity to errors in transfer orbits for Hohmann 
transfers is given in [Kamel and Soliman, 2006]. This may be directly applied to 
tethers for the same type of manoeuvres. For example, a 0.1% error in 
V 
may 
give up to a 5% error in radius and semi-major axis. A thorough treatment of the 
errors due to an early or late release is needed for the tether. 

Intermsofreleasestrategyforan inclined orbit,thiscriteriaforrelease isgenerally 
only satisfied at one point in the cycle when the inclined orbital plane cuts the 
inertialplane,thedirection isdeterminedfromtheclassicalHohmanntrajectory,as 
in [McLaughlin, 2000]. 

An alternative to the Hohmann trajectory is the WSB method, which has been 
describedinthe literature:[Belbruno,1987],[MillerandBelbruno,1991] and[Koon 

9Thisassumesthatthemission istoprovidethemaximum
V 
;many alternativetether missions 
exist, including a inclinationchangeusing atetherreleasing thepayload atanangletotheorbital 
plane. 

86 


et al., 2001]. To summarise, the WSB utilises the gravitational interaction of the 
Earth-Sun-Moon system to lower the 
V 
required to capture a satellite in Lunar 
orbit. The WSB can be thought of as two Circular Restricted 3 Body Problems 
(CR3BP)patched together with a small correction burn of less than 25m/s 
at the 
boundary between the Sun-Earth system and the Earth-Moon system. 

A patch point is located by integrating forward the dynamics of the spacecraft 
in the Earth-Moon system from the patch point until the spacecraft is ballistically 
captured in Lunar orbit, and simultaneously integrating backward the dynamics of 
the spacecraft in the Sun-Earth system until the spacecraft links up with a LEO 
orbit around the Earth. Thus a patch point is found along with two halves of the 
trajectory from LEO to ballistic capture in Lunar orbit via the patch point. 


Figure4.6: Diagram showing an example of the WSB trajectory, fromESABulletin 

103: [Biesbroek and Janin, 2000] 
Timing the release of the tether payload to launch on a WSB transfer is a difficult 
computational puzzle, but one which may be solved by optimising a numerical 
solution to the equations of motion. A trajectory is planned for a future time and 
the tether mission must provide a 
V 
precisely with a very small margin of error 
in release angle. 

The useful tip velocity(
V 
)willbeapproximatelyequal tothe localbody-centred 

87 



velocity of the tip payload. The local velocity is given in Equation 4.44 below: 

.. 


.

-L 
sin()cos( ).-L 
cos()sin( ). 


.. 


.. 


Vpayload 1 = .-L 
sin()sin( ).+L 
cos()cos( ) .
(4.44) 

. 
. 
. 
. 
-L 
cos().a 


This reduces to the simple one-dimensional angular velocity when considering the 
optimal release case of a perfectly aligned tether of . 
. 0, a 
. 0 and .a 
. 0: 

.. 


0 

.. 


.. 


..

Vpayload 1 . L .
(4.45) 

.. 


.. 


0 

The major long-term perturbations such as J2, Lunar and Solar gravity etc. are 
not covered here as they do not affect the short-term dynamics of tether spin-up. 
[Parsons, 2006] covers some of these alongside the WSB trajectories. 

4.4.1 Tether error analysis 
Small changes in the tether variables could have a large effect on the performance 
of the tether, therefore it is important to quantify these for the purposes of risk 
mitigation. Ananalysiswasperformed toquantify theeffectsthat asmallchange in 
thesystemparametershave onthepayload apogeeand inclination, whencalculated 
at the orbital location directly opposite the separation point10 . 

The Figures 4.7 and 4.8 are calculated by solution of the equations of motion for 
MMET tether system with two payloads, with the default initial conditions: 

Lpay 
=1000m, 
Mpay 
=10kg, 
Mtether 
=100kg, 
Mhub 
=500kg, 
e 
=0, 
..

.[0] =0rad/s, 
 [0] =0.1rad/s, 
.[0] =0rad/s, 
R[0] =0m/s, 


10Calculatedas p 
radians 
roundthe orbitfrom the system centre of masspoint when thepayload 
is separated from the tether. 

88 


.[0] =0.00113905rad/s, 
[0] =29/180rad, 
 [0] =0rad, 
[0] =0rad, 
R[0] =0m, 
[0] =0rad, 
Torque =1000Nm, 
NominalOrbitalPeriod =6306.95s. 


The initial conditions above equate to an equivalent 
V 
of 100m/s. 

Thepayload isreleased whenthetether isaligned withtheradiusvector(. 
=0 
and a 
= 0) and follows a ballistic trajectory thereafter. No allowance is made here 
for angular momentum effects between the payload and tether system on release. 

Errors are simulated by the addition of small offsets in the initial conditions at 
the release point. The effect that the errors have on that variable is analysed by 
comparing the apogee and the inclination to the unperturbed values measured at the 
apogee point. The starting radius11 is 7, 
378, 
000m. With the default initial conditions 
above, the apogee is 7, 
747, 
377m, a difference of 369, 
377m 
or approximately 
5% of the radius vector. In terms of altitude, the starting altitude is 100, 
000m 
and 
theapogeewhenthepayloadisreleased is469, 
377m. The inclination of the apogee 
point is equal to the magnitude of the perigee inclination of 29. . 

Figure4.7ashowstheeffectthatasmallchange inradiushasonthetetherapogee. 
A change in radius of 1000m 
at the tether release point corresponds to a 7460m 
increase intheapogeeof thepayload. Thedifferenceoftheapogee ishigherthanthe 
expected first order approximation of 7
L 
as suggested12 in [Ziegler and Cartmell, 
2001]by 460m, the higher calculated number is a result of the 
V 
supplied by the 
rotating tether. 

The change in apogee due to small changes in inclination is shown in Figure 4.7b. 
It follows from orbital mechanics that a small change in the inclination to flatten 
the orbit will increase the apogee. 

A small change in the in-plane rotation angle,  , causes minor changes to the apogee 
altitude, as shown in Figure 4.7c. The relationship is predominantly parabolic 

11That is, the perigee point. 
12For a hanging tether. 


89 


based due to the increase in energy supplied to the orbit 12m
V 
2, which is itself 
dependent on  .
. 

4.4.2 Lunar capture 
Once the spacecraft is ballistically captured around the Moon, a small thrust is 
required to ensure that the orbital energy is reduced below the threshold required 
for the spacecraft to escape. A 
V 
of less than 50m/s 
will fix the spacecraft in a 
highly elliptical(e 
=0.9) but stable Lunar orbit. 

A midpoint burn of 
V 
= 25m/s 
with a capture burn of 
V 
= 50m/s 
corresponds 
to a fuel mass fraction of 9.6% assuming an Isp 
rating of 75s 
for a Nitrogen 
coldgas thruster. Aconservative estimate of1kg 
massfor the engine and associated 
structure leavesthe10kg 
original satellite mass with8.04kg 
of usefulpayload mass. 

When this is compared to the Hohmann transfer 
V 
cost, [Koon et al., 2001,p71] 
calculates that the WSB transfer requires 20% less fuel to execute. This necessarily 
transfers into a lower mass of the payload and subsequent cost savings. 

Once the payload is captured, it is possible to use tethers for the descent to the 
Lunarsurface. Staged tethersas in[Cartmellet al.,2004]orLunarelevatorsmaybe 
used to transfer the payload to the surface, and Lunar minerals to LEO. [Williams 
et al., 2005, p 784] claims that the payload can be delivered within 1cm 
for a 
controlled handover of a staged tether system, and can be held in a 10m 
window 
for approximately 135 seconds, which is ideal for the multi-staged tether system 
provided a control system can be devised that will enable an automatic remote 
handover of payloads. 

90 



4.5 
A demonstration Lunar mission with a tether 
launch 
A demonstration mission to the Moon is outlined below, involving a tether launch 
fromLEO toLunarparking orbit. It isexamined using hardwareavailable in2008, 
as a low-cost micro-satellite mission that could be launched in the short term, with 
today’s technology. 

4.5.1 
Spacecraft sizing 
The MMET system is sized to launch a 10kg 
payload to Lunar capture orbit. This 
is intended as a demonstration level mission, although the MMET system may be 
scaled to launch larger payloads, as discussed previously. 

The facility is proposed to be entirely housed within a cylinder containing the 
motor andgearbox. The solar arrays will consist of35000 standard sized2cmx4cm 
solar cells, whichprovide thepower requiredbythe motor, rechargethebatteries and 
providepower to the spacecraftbus. The solar cells willbe aInGaP/GaAs/Ge 
type 
capable of 25% efficiency [Green et al., 2003], with a degradation rate of 1%/year 
[GriffinandFrench,1991].Themission length of5years isbased ontheprobability 
of a tether cut due to orbital debris when using multiple redundant tether strands, 
coupled with the useful life of the solar arrays. 

Thepowertothemotorwillonlybeavailableduring thetimethefacility isnot in 
eclipse and this is factored into the equations using the function Eclipse described 
in Section 4.2.5. 

The totalusefulpower availablefor a2mx3m 
cylinder augmentedby a1.5mx6m 
fold out panel will be 4.3kW 
when launched and 3.05kW 
atthe end of a5year 
mission. The end of life power allows the motor characteristics to be defined, in 
this case the motor used will be a 3kW 
rated motor capable of driving 1000Nm 


91 



of torque through a reducing gearbox. The gearbox will reduce the rotations from 
the motor armature rotation of 430rad/s 
to the required angular velocity of the 
rotor arm of 20rad/s. This motor is based around GE Industrial DC motor model 
5BC49JB1115, rated at 4hp 
and 4150RP 
M, which is designed to drive an electric 
vehicle. 

The rotor is comprised of two 1km 
tether sections with each linking the facility 
to the payloads. The stator is identical, with the payloads replaced by counterweights. 
The tether itself is Zylon. 
fibre with a tensile strength of 5.8GP 
a, density 
of 1570kg/m3 and cross sectional area of 64mm2 . This leads to a tether mass of 
100kg 
for each of the four tether lines, totalling 400kg. A safety factor of 1.3 is 
applied to the calculation of the maximum breaking stress, however additional precautions 
must be taken by careful design of the tether structure to ensure the tether 
will remain intact if impacted by micro-meteorites or debris [Forward and Hoyt, 
July 1995]. 

The tether mass outweighs the payload mass in order to keep the breaking stress 
to a manageable level. This is advantageous as the tethers may be adjusted to 
compensate for any tether failure mode or asymmetric payload release. 

The MMET is designed to be highly reusable. The 5 day spin up cycle must be 
matched by an equal 5 day spin down cycle to return the rotor to a stationary 
positionaligned with thegravity vector. Whenthis isachieved,thepayload willbe 
much easier to dock than when in motion. For the WSB transfer, payloads can be 
transferred to the Moon every Lunar month, leaving a window of 20 days to dock 
the payloads to the tethers. 

92 



4.5.2 Analysis 
The square of the characteristic velocity, Uvel, of the tether is proportional to the 
tensile strength, T, and inversely proportional to the density, : 

2T 


Uvel = (4.46) 

. 


The characteristic velocitydeterminesthe maximum velocityattainablefor atether 
supporting its own mass while rotating. Adding a payload limits the velocity a 
tether can impart by increasing the tension and stress in the tether, therefore by 
keeping the mass of the payload to a fraction of the tether’s mass, the decrease in 
characteristic velocity is minimised. 

A tether manufactured from Zylon. 
with T 
=5.9GP 
a, . 
= 1570kg/m3 with a 
safety factor of 1.3 gives a characteristic velocity of 2.238km/s. With a payload 
mass equivalent to 10% of the tether mass attatched to the end of the tether, the 
characteristic velocity will drop by approximately 10% to 2.10km/s. To achieve 
Lunar orbit, thepayloadwill need3.187km/s 
[Sweetster,1991]fromLEO at167km 
altitude. Therefore, the tether will have to occupy a higher earth orbit of 1000km 
to successfully launch to the moon, or a staged system will be required [Cartmell 
et al., 2004]. 

The necessary orbital altitude of 1000km 
is unfortunate in terms of the orbital 
debris that the tether will encounter. Figure 4.9 shows the debris environment from 
LEO to Geostationary Earth Orbit (GEO) – the debris environment is relatively 
dense until 3000km 
altitude, meaning the risk of a severed tether is higher in this 
region of space. This is compounded by the long time to clear the debris from the 
higher orbits, meaning the orbitaldebris will continuetoposeproblemsfordecades 
to come. 

93 



4.6 Conclusions 
A demonstration mission launching a micro-satellite from MEO to Lunar orbit is 
achievable using current technology. The safety margins, however, are extremely 
small and the MMET launcher would be located in an orbit with large amounts of 
debris. 

The inclination term does not significantly alter the dynamics of the MMET system. 
However,anyout-of-planecomponentinthelocal axes –whetheronaninclined 
orbitornot –interactswith theorbitalparameterstocreateanunsteady oscillation 
in the tether. This is unacceptable for payload aiming and release, therefore the 
local out-of-plane angle should be minimised insofar as possible. 

94 



Apogee change @m. 
Apogee change @%. 


-1000 -500 500 1000 
-6000 
-4000 
-2000 
2000 
4000 
6000 
Rad 


-0.01 -0.005 0.005 0.01 
-0.01 
-0.005 
0.005 
0.01 
Inc 


change 


change 


@%D

@m. 


(a) Payload apogee change (m) vs. change (b) Payload apogee change (%) vs. change 
in radius (m). Solid line – calculated in inclination (%). Solid line – calcuresults, 
dotted line – linear relationship lated results, dotted line – linear relationship 

Apogee 
=7* 
R 

Apogee 
= -1* 
. 
-1.5 -1 -0.5 0.5 1 1.5 
-175 
-150 
-125 
-100 
-75 
-50 
-25 
Apogee change @m. 


Apogee change @m. 


Psi 
@deg. 


-0.01 -0.005 0.005 0.01 
-30000 
-20000 
-10000 
10000 
20000 
30000 
PsiDot 
@rad�s. 


(c) Payload apogee change (m) vs. change (d) Payload apogee change (m) vs. change 
.
in . 
(deg). Solid line – calculated res-in . 
(rad/s). Solid line – calculated 
ults, dotted line – trigonometric relationship results, dotted line – linear relationship 

Apogee 
=((
 ))2 
*-60 
Apogee 
=3500000* 
 .

Figure 4.7: Change in apogee of the payload after release from the tether system 
after one half of an orbit compared to small changes of system parameters 

95 



Apogee change @m. 
Apogee change @m. 


-1 -0.5 0.5 1 
-3000 
-2000 
-1000 
1000 
2000 
3000 
-10 -5 5 10 
-4000 
-2000 
2000 
4000 
Length


Alpha 


Change


@Deg. 


@m. 


(a) Payload apogee change (m) vs. change (b) Payload apogee change (m) vs. change 
in a 
(deg). Solid line – calculated in length (m). Solid line – calculated 
results, dotted line – linear relationship results, dotted line – linear relationship 

Apogee 
= -3000* 
a 

Apogee 
=400
L 
Inclination change @%. 


-1 -0.5 0.5 1 
Alpha 
@degD
-0.0004 
-0.0002 
0.0002 
0.0004 
(c) Payload inclination change (%) vs. 
change in a 
(deg). Solid line – calculated 
results, dotted line – linear relationship 

. 
=

- 1 


2000 


Figure 4.8: Change in apogee and inclination of the payload after release from 
the tether system after one half of an orbit compared to small changes of system 
parameters 

96 



Figure 4.9: The orbital debris environment from LEO to GEO, from [Wikipedia, 
2008] 

97 



4.7 Inclination plots 
Table 4.3 shows the varying cases with an explanation of their initial conditions. 
This is an exact duplicate of Table 4.2, reprinted for reference. 

.

case .  .. 
a 
eccent t 


I1 0 0 0 0 0 0 0 1000 
I2 0 0 0 29/180 0 0 0 1000 
I3 0 1 0 29/180 0 0 0 1000 
I4 0 0 0 29/180 0 -29/180 0 1000 
I5 0 1 0 29/180 0 -29/180 0 1000 

Table 4.3: Initial conditions for the inclination numerical solutions. 

The initial conditions for the other parameters held constant are: 
L 
=1000m, 
Mpayload =10kg, 
Mtether =100kg, 
Mfacility =500kg, 
e 
=0, 
R.
[0] =0,.
[0] =0.00113905rad/s, 
[0] =0,R[0] =7.378* 106 m 


98 



Orbit Radius 


Orbit Radius 


7


1.4·10



6


7 


7.378·10


1.2·10


7


1·10


6


7.378·10


6


8·10


6


6 


7.378·10


6·10


6


4·10


6


7.378·10


6


2·10



t



t 


21600 43200


21600 43200 


(a) Case 1: R 
(m)vs.time(s) (b) Cases 2-5: R 
(m)vs.time(s) 
TT 


60 



60 


50 


50 



40 



40 


30 


30 


20 


20 


10 


10 



t 


t 


21600 43200 


21600 43200 



(c) Case 1: . 
(rad)vs.time(s) (d) Cases 2-5: . 
(rad)vs.time(s) 
inclination inclination 


1 


0.5 


t 


21600 


43200 


-0.5 


-1 


21600 43200 
t 
-0.4 
-0.2 
0.2 
0.4 
(e) Case 1: . 
(rad)vs.time(s) (f) Cases 2-5: . 
(rad)vs.time(s) 
Figure 4.10: Plots of R, . 
and . 
vs. time for the five cases. 
99 


21600 43200 
t 
-1 
-0.5 
0.5
1
. 
(a) Case 1:  (rad) vs. time (s) 
21600 43200 
t 
-0.2 
-0.1 
0.1 
0.2
. 
(b) Case 2:  (rad) vs. time (s) 
21600 43200 
t 
-0.0015 
-0.001 
-0.0005 
0.0005 
0.001 
0.0015
. 
(c) Case 3:  (rad) vs. time (s) 
21600 43200 
t 
-0.5 
-0.4 
-0.3 
-0.2 
-0.1 
0.1
. 
(d) Case 4:  (rad) vs. time (s) 
21600 43200 
t 
-0.4 
-0.2 
0.2 
0.4
. 
(e) Case 5:  (rad) vs. time (s) 
Figure 4.11: Plots of  vs. time for the five cases. 
100

21600 43200 
t 
-1 
-0.5 
0.5
1 
D@.,tD 
(a) Case 1: . (rad/s) vs. time (s) 
21600 43200 
t 
-0.01 
-0.005 
0.005 
0.01 
D@.,tD 
(b) Case 2: . (rad/s) vs. time (s) 
21600 43200 
t 
-0.0015 
-0.001 
-0.0005 
0.0005 
0.001 
0.0015 
D@.,tD 
(c) Case 3: . (rad/s) vs. time (s) 
21600 43200 
t 
-0.004 
-0.002 
0.002 
0.004 
D@.,tD 
(d) Case 4: . (rad/s) vs. time (s) 
21600 43200 
t 
-0.3 
-0.2 
-0.1 
0.1 
0.2 
0.3 
D@.,tD 
(e) Case 5: . (rad/s) vs. time (s) 
Figure 4.12: Plots of . vs. time for the five cases. 
101

-1 -0.5 0.5 1 
i 
-1 
-0.5 
0.5
1
. 
(a) Case 1:  (rad) vs.  (rad) 
-0.4 -0.2 0.2 0.4 
i 
-0.2 
-0.1 
0.1 
0.2
. 
(b) Case 2:  (rad) vs.  (rad) 
-0.4 -0.2 0.2 0.4 
i 
-0.0015 
-0.001 
-0.0005 
0.0005 
0.001 
0.0015
. 
(c) Case 3:  (rad) vs.  (rad) 
-0.4 -0.2 0.2 0.4 
i 
-0.5 
-0.4 
-0.3 
-0.2 
-0.1 
0.1
. 
(d) Case 4:  (rad) vs.  (rad) 
-0.4 -0.2 0.2 0.4 
i 
-0.4 
-0.2 
0.2 
0.4
. 
(e) Case 5:  (rad) vs.  (rad) 
Figure 4.13: Plots of  vs.  for the five cases. 
102

-0.00001 
-5·10-6 
0 
5·10-6 
0.00001 
.'' 
0 
0.1 
0.2 
0.3 
0.4 
.' 
0 
5000 
10000 
.
(a) Case 1:  , . 
, ¨   phase plot 
-0.0001
-0.00005 
0 
0.00005 
0.0001 
.'' 
0 
0.1 
0.2 
0.3 
0.4 
.' 
0 
5000 
10000 
. 
(b) Case 2:  , . 
, ¨   phase plot 
0 
5·10-6 
0.00001 
.'' 
1 
1.1 
1.2 
1.3 
1.4 
.' 
0 
20000 
40000 
60000 
. 
(c) Case 3:  , . 
, ¨   phase plot 
-0.00002 
0 
.'' 0.00002 
0 
0.1 
0.2 
0.3 
0.4 
.' 
0 
5000 
10000 
. 
(d) Case 4:  , . 
, ¨   phase plot 
-0.05 
0 
.'' 0.05 
1 
1.2 
1.4 
.' 
0 
20000 
40000 
60000 
. 
(e) Case 5:  , . 
, ¨   phase plot 
Figure 4.14: Phase plots of   for the five cases. 
103

-1 
-0.5 
0 
0.5 
1 
.'' 
-1 
-0.5 
0 
0.5 
1 
.' 
-1 
-0.5 
0 
0.5 
1 
. 
(a) Case 1: , . , ¨ phase plot 
-0.002 
0 
.'' 0.002 
-0.01 
0 
0.01 
.' 
-0.2 
0 
0.2 
. 
(b) Case 2: , . , ¨ phase plot 
-0.001 
0 
.'' 0.001 
-0.001 
0 
.' 0.001 
-0.001 
0 
0.001 
. 
(c) Case 3: , . , ¨ phase plot 
-0.001 
0 
.'' 0.001 
-0.005
-0.0025 
0 
0.0025
0.005 
.' 
-0.4 
-0.2 
0 
. 
(d) Case 4: , . , ¨ phase plot 
-0.2 
0 
.'' 0.2 
-0.2 
0 
0.2 .' 
-0.5 
-0.25 
0 
0.25 
0.5 
. 
(e) Case 5: , . , ¨ phase plot 
Figure 4.15: Phase plots of  for the five cases. 
104

21600 43200 
t 
100 
200 
300 
400 
DeltaV 
(a) Case 1: 
V (m/s) vs. time (s) 
21600 43200 
t 
100 
200 
300 
400 
DeltaV 
(b) Case 2: 
V (m/s) vs. time (s) 
21600 43200 
t 
1100 
1200 
1300 
1400 
DeltaV
(c) Case 3: 
V (m/s) vs. time (s) 
21600 43200 
t 
100 
200 
300 
400 
DeltaV 
(d) Case 4: 
V (m/s) vs. time (s) 
21600 43200 
t 
1000 
1100 
1200 
1300 
1400 
DeltaV
(e) Case 5: 
V (m/s) vs. time (s) 
Figure 4.16: Plots of 
V vs. time for the five cases. 
105

21600 43200 
t 
2000 
4000 
6000 
8000 
10000 
TetherTension N 
(a) Case 1: Tension (N) vs. time (s) 
21600 43200 
t 
2000 
4000 
6000 
8000 
10000 
TetherTension N 
(b) Case 2: Tension (N) vs. time (s) 
21600 43200 
t 
70000 
80000 
90000 
100000 
110000 
120000 
TetherTension N 
(c) Case 3: Tension (N) vs. time (s) 
21600 43200 
t 
2000 
4000 
6000 
8000 
10000 
TetherTension N 
(d) Case 4: Tension (N) vs. time (s) 
21600 43200 
t 
60000 
70000 
80000 
90000 
100000 
110000 
TetherTension N 
(e) Case 5: Tension (N) vs. time (s) 
Figure 4.17: Plots of Tension vs. time for the five cases. 
106

21600 43200 
t 
2.5·107 
5·107 
7.5·107 
1·108 
1.25·108 
1.5·108 
Tether Stress HPa L 
Tether Intact 
(a) Case 1: Stress (N/m2) vs. time (s) 
21600 43200 
t 
2.5·107 
5·107 
7.5·107 
1·108 
1.25·108 
1.5·108 
Tether Stress HPa L 
Tether Intact 
(b) Case 2: Stress (N/m2) vs. time (s) 
21600 43200 
t 
1.2·109 
1.4·109 
1.6·109 
1.8·109 
Tether Stress HPa L 
Tether Intact 
(c) Case 3: Stress (N/m2) vs. time (s) 
21600 43200 
t 
2.5·107 
5·107 
7.5·107 
1·108 
1.25·108 
1.5·108 
Tether Stress HPa L 
Tether Intact 
(d) Case 4: Stress (N/m2) vs. time (s) 
21600 43200 
t 
1·109 
1.2·109 
1.4·109 
1.6·109 
1.8·109 
Tether Stress HPa L 
Tether Intact 
(e) Case 5: Stress (N/m2) vs. time (s) 
Figure 4.18: Plots of Stress vs. time for the five cases. 
107

Chapter 5 

Dynamics of tethers with length 
deployment 

Ashasbeenshown inChapter4, it ispossibletodescribethemotionoftheMMET 
around complex orbital paths. Now the deployment characteristics of the MMET 
will be explored. 

5.1 Deployment and recovery with the MMET 
Thetwopayload armsof theMMET maybothbegainfullyemployed to launch two 
payloads,andthispresentsaset of uniquechallengesforthe launcher. Bothpayload 
tethers must be able to demonstrate all the necessary properties of a single system 
tether, with the additional demands of a dual-launch system overlaid. The tethers 
must be strengthened for a potential asymmetric release, and the oscillations that 
this imposes on the system and tethers. 

Thegeneralised coordinateshavebeenchosen inordertoascertainthe impactthat 
the length deployment has on the MMET without over-complicating the dynamics 

108 



of the system. The global coordinates R 
and . 
have been included to model the 
orbital dynamics. The primary rotational coordinate,  , has been expanded to 
include both payload arms in the form of  1 and  2. Similarly, each payload arm 
has been assigned a length coordinate, L1 and L2, to enable independent, rigid 
motion of the extendable tether arm. 

To limitthenumberofgeneralised coordinates,theinclinationvariable, , and the 
out-of-plane coordinate, , have not been included as generalised coordinates. A 
six degree of freedom model will be adequate to examine the dynamics of length 
deployment,howeveraninedegreeoffreedommodelhasthepotential to introduce 
orbital resonance to the deployment. At this early stage of research, nine degrees 
of freedom are too many to successfully isolate and examine the effects of length 
deployment on the system. 

5.2 Additionoflength asageneralised coordinate 
The six variables of interest – {R, 
, 
 1, 2,L1,L2}– fulfil all the requirements of 
generalised coordinates. Ananalysiswillbeperformed to investigatetheeffectthat 
deployment and recovery have on the tether system orbit. 

A system of equations will be constructed to include a symmetrical dumbbell 
tether with a facility mass at the centre of the system. The important mass points 
included in the model are the two identical payload masses, Mpayload, the facility 
mass, Mfacility, and the n 
discretized tether masses, Mtether, which are evenly distributed 
along each tether. 

As before, the stator is not included in this model because this would significantly 
increase the complexity of the system. 

As previously, Lagrange’s equations will now be derived for the system. 

109 



5.3 Constructing the equations of motion 
A system of Lagrange’s equations is constructed using the rotational method described 
in Chapter 3, similarly employed for the inclination equations in Chapter 4. 

The localpositionof themasseswith respecttothefacility massarederived. This 
allows the centre of mass with respect to the facility mass to be calculated. The 
orbital positions are then assembled from these equations. 

Simplifying assumptions have been made to lower the computational cost of assembling 
and solving these equations. 

. 
Theorbitallength used inthecalculationofpotential energy isassumed tobe 
the orbital radius vector, R. This is acceptable as L 
« 1. 

. 
R 


The orbital rotations have been decoupled from the local rotations. This is 
sufficient to analyse the fast rotations of the dual-payload system, but should 
be treated with caution when analysing the stationary characteristics of the 
MMET. 

. 
The tethers are assumed to be rigid and inelastic. 

. 
Thecross-sectional area(CSA) of thetether isassumedtobeconstantalong 
thetether length. TheCSA willalmost certainlybeafunctionoflength when 
thetether isbuilt(i.e.tapered),however,thisassumptionbuilds ina level of 
conservatism into the analysis which can be refined at a later stage. 

These assumptions aredesigned tomaximisethebenefit of thesolutiontotheequations, 
while minimising the computational cost and minimising the risk of losing 
that information while studying a multi-parameter system. 

As outlined in the above assumptions, the orbital and local rotations have been 
decoupled. The positional equation of the first payload therefore is defined as: 

Pfinal1 = R,Z · (P0.CoM)-(Pfacility.CoM)+(R 1,Z · P1) (5.1) 

110 


Ascanbeseen inEquation5.1,therotationsarepurelybased onaseriesofZ-axis 
rotations. 

Using the rotation matrices defined in Section 3.5, along with the following definitions1 
for the payload position: 

.. 


L1 

.. 


.. 


..

P1 = 0 (5.2) 

.. 


.. 


0 

leads to the expanded version of the positional equation for payload 1: When 
evaluated, this becomes: 

..... 
. 


R 
cos . 
CoMX 
L1 cos( 1) 

..... 
. 


..... 
. 


..... 
.

Pfinal1 = R 
sin. 
- CoMY 
+ L1 sin( 1) (5.3) 

. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
0 CoMZ 
0 

Thepositionof thecentre of massfromthefacility, P0.CoM, is calculated with the 
system mass points including the payload, facility and tether mass points. 

Thetether isdiscretized with each tethercomprising n 
equal masspoints, as shown 
in Equation 5.4 and Figure 5.1. The aim in discretizing the tether is to enhance 
the potential and rotational energy expressions. The n 
discrete mass elements were 
chosen sothecentre of mass of thetetherfallsonthegeometric centre of thetether, 
necessitating n 
+1 spaces between the elements. 

.. 


n

n

(M1+Mtether 1)L1 cos( 1)+(M2+Mtether 2)L2 cos( 2)

22 

.. 


M1+M2+Mfacility+nMtether 1+nMtether 2 

n

n

(M1+2 Mtether 1)L1 sin( 1)+(M2+2 Mtether 2)L2 sin( 2) 

. 
M1+M2+Mfacility+n 
Mtether 1+n 
Mtether 2 . 
. 
. 
0 

. 


. 


P0.CoM 

(5.4) 
.

= 

. 


To create Lagrange’s equations, the position, velocity and energy of each point 

1Where L1 
is the length in absolute terms between the facility and the payload. 

111 


Figure 5.1: Discretised tether diagram with n= 5 mass points Figure 5.1: Discretised tether diagram with n= 5 mass points 
must be determined. 
The positions of all the mass points are given as: 


..... 
. 


R 
cos. 
CoMX 
L1 cos( 1) 

..... 
. 


..... 
. 


..... 
.

Ppayload 1 = R 
sin . 
- CoMY 
+ L1 sin( 1) (5.5) 

..... 
. 


..... 
. 


0 CoMZ 
0 

..... 
. 


R 
cos. 
CoMX 
L2 cos( 2) 

..... 
. 


..... 
. 


..... 
.

Ppayload 2 = R 
sin . 
- CoMY 
+ L2 sin( 2) (5.6) 

..... 
. 


..... 
. 


0 CoMZ 
0 

..... 
. 


R 
cos. 
CoMX 
L1 cos( 1) 

..... 
. 


..... 
. 
j 


..... 
.

Ptether 1,j = R 
sin . 
- CoMY 
+ L1 sin( 1) (5.7) 

..... 
. 


n 
+1 

..... 
. 


0 CoMZ 
0 

..... 
. 


R 
cos. 
CoMX 
L2 cos( 2) 

..... 
. 


..... 
. 
j 


..... 
.

Ptether 2,j = R 
sin . 
- CoMY 
+ L2 sin( 2) (5.8) 

..... 
. 


n 
+1 

..... 
. 


0 CoMZ 
0 

. 
.. 
. 


R 
cos. 
CoMX 


. 
.. 
. 


. 
.. 
. 


. 
.. 
.

Pfacility = R 
sin . 
- CoMY 
(5.9) 

. 
.. 
. 


. 
.. 
. 


0 CoMZ 


where each j 
is the number of the n 
tether mass points. 

The velocity of each point is determined. The payload 1 is used as an illustrative 

112 



example: in local axes in Equation 5.10 and in inertial axes in Equation 5.11 

.. 


L. 1 cos( 1)-L1 sin( 1) . 1 

.. 


.. 


..

Vpayload1,local = L. 1 sin( 1)+L1 cos( 1) . 1 (5.10) 

.. 


.. 


0 

Vpayload1,inertial = 

... 
. 


R.
cos()-R. sin() -L1  . 1 sin( 1)+L. 1 cos( 1) 

... 
. 


... 
. 


. 
.
.. 
.

R 
sin()+R. cos()+ L1  . 1 cos( 1)+L. 1 sin( 1)

... 
. 


... 
. 


00 

1

+ · 
M1+M2+Mfacility+2nMtether 

.. 
. 


n 
(5.11) 

- M1 + Mtether L1  . 1 sin( 1)-L. 1 cos( 1) 

.. 
2 
. 


.. 
. 


.. 
n 
. 


.. 
- M1 + 2 Mtether L1  . 
1 cos( 1)+ L. 1 sin( 1) 
. 


.. 
. 


0 

. 
.. 


n

M2 + 2 Mtether L2  . 2 sin( 2)-L. 2 cos( 2) 

. 
.. 


. 
.. 


. 
n 
..

+ 
. 
M2 + 2 Mtether L2  . 2 cos( 2)-L. 2 sin( 2) 
.. 
. 
.. 


0 

The expressions for energy are then calculated from the velocity and position 

information. 

The translational kinetic energy, Ttrans, of the system is calculated for the MMET 

113 



tether with n 
discrete masses per tether2: 

M1 M2
Ttrans = Vpayload 1 · Vpayload 1 + Vpayload 2 · Vpayload 2 


2 
2 

n 


Mtether 

+ Vtether 1j · Vtether 1j 
2 

j=1 
n 


Mtether 

+ Vtether 2j · Vtether 2j 
2 

j=1 

1 

+(M1 + M2 + nMtether + nMtether + Mfacility)(VCoM · VCoM) (5.12) 
2 

The rotational kinetic energy, Trot, of the system is found in a similar way: 

Trot =(Ipayload 1 + Ipayload 2)· (!local · !local) 

n 
n 


X. 
(5.13) 

+ 
Itether 1j · (!localj · !localj)+ Itether 2j · (!localj · !localj) 
j=1 j=1 
where: 

.. 
.. 


0 
100 

.. 
.. 


.. 
.. 


.. 
..

!local =0 
I3 = 010 

.. 
.. 


.. 
.. 


.

. 
001 

=L2 
=L2

Ipayload 1 1Mpayload 1 · I3 
Ipayload 2 2Mpayload 2 · I3 

Itether 1j =L21j 
Mtether 1j · I3 
Itether 2j =L22j 
Mtether 2j · I3 

The evaluated sum of the component rotational kinetic energies for n 
=10 tether 

masses are: 

22 

L12 M1  . 1 L22 M2  . 2 35Mtether 22 2 

Trot =++ L1  . 1 + L22  . 2 (5.14) 

2 222 

2Where Mtether 
is the discrete mass point, this sums to the total tether mass M 
= nMtether. 

114 



The sum of the potential energies, U, are: 

n 


µmj

U 
= 
. 
(5.15) 

j=1 Pj 


where Pj 
are the mass point locations given in Equations 5.5 to 5.9. 

The potential energy term is simplified by using the assumption that the mass 
points are sufficiently close to the centre of mass of the system, that is PR 
j 
« 1, that 
the potential energy of each mass point of the system may be represented as if it 
were at a distance R 
fromthe majorgravitationalbody. The evaluatedform of one 
tether arm is therefore: 

U 
= 
n 
. 
µmj 
R 
= 
n 
Mtether 
R 
(5.16) 
j=1 

Summing over all the mass points3 gives the full form for the potential energy of 
the MMET: 

µM1 µM2 20µMtether µMfacility 


U 
= --- - (5.17) 

RRR 
R 


The Lagrangian is then given by L 
= T 
-U. 
Lagrange’s equations are then generated for the five generalised coordinates, 


{R, 
, 
 1, 
 2, 
L1, 
L2}as follows: 
. 
. 
d 
@T 
@T 
@U 
dt 
@q.j 
- @qj 
+ 
@qj 
= Qj 
(5.18) 

5.4 Non-conservative forces 
The motor on the MMET spins up the payload arms, while the counter-torque 
rotates the stator in the opposite direction. This is a non-conservative force in 
action, as the energy is transferred from the solar cells4 to the rotational motion of 

3There are 10 discrete mass points on each tether. 
4Or the equivalent driving power source. 


115 


the payloads, via the motor. The non-conservative forces are derived according to 
the principles of virtual work outlined in Section 3.4: 

.. 


0 

.. 


.. 


..

0 

.. 


.. 


.. 


. 
t 
cos(
) 
. 


..

Q 
= (5.19) 

.. 


. 
t 
cos(
) 
. 


.. 


.. 
2 . 


.. 


. 
- Ldamp 
L. 1 
. 


..

. 
2 . 


- Ldamp 
L. 2 

where t 
is the torque magnitude, . 
is the angle that the torque acts relative to 
the in-plane spin and Ldamp 
is a braking term modelling the action of the spool-out 
mechanism. The form of the tether friction braking was taken after the velocity 
based barber-pole braking system analysed in [Lennert and Cartmell, 2006]. 

5.4.1 Tether braking 
It is proposed to use a barber pole braking system for this tether analysis. This 
is a passive system employed to minimise the need for an active control system 
or complicated reel-out mechanism. The kinetic energy of the tether and payload 
spoolsoutthetetherwithfrictionforcesproviding resistancetoensurethetether is 
deployed in a controlled manner. 

The braking force on the tether will be proportional to the square of the speed of 
deployment: 

L.
2

F 
= - Ldamp 
(5.20) 

where Ldamp 
is aproportional constant modellingthefrictionproperties andphysical 
characteristics of the braking system. This allows a flexibility to either define the 
damping coefficient to reflect a physical braking system or to specify a damping 
coefficient and match a braking system to the numerical modelling. 

116 


5.5 Analysis of length deployment of tether 
The MMET is different to other tether systems by virtue of the dual-payload motorized 
arrangement of its tethers. This confers several advantages through the 
symmetric arrangement of the payloads, however the dynamics of the spin-up are 
untested in an orbital environment5 . 

A goal is set to study the dynamics of the MMET system: it must be capable of 
imparting a
V 
=1000m/s 
tothepayloads. This isamoderatevelocity increment 
that will demonstrate the usefulness of the MMET system, allowing inclusion on a 
small launcher or as a secondary payload. 

The system of equations is solved in Mathematica. 
, using NDSolve, for a time 
of t 
= 208129s 
(approx 33 orbits). Figures 5.2 and 5.3 show the results for the 
following initial conditions: 

100 

M1 = M2 =10kg, 
Mtether = 10 =10kg, 
Mfacility =500kg, 


R.
[0] =0,.
[0] =0.00113905rad/s, 
 . 1[0] =0.1rad/s, 
 . 2[0] =0.1rad/s, 


L. 1[0] =0,L. 2[0] =0,R[0] =7.378* 106 m, 
[0] =0, 1[0] =0.01., 2[0] =180.1. , 


L1[0] =100m, 
L2[0] =100m, 
Ldamp 
=5* 108 kg/m, 
max =1000Nm 


A further explanation of the starting conditions is below: 

. 
the starting radius R[0]isequivalentto1000km 
abovethe surface oftheEarth. 

. 
the motor torque is proportional to time, and increases from t 
=0 at t 
=0s 
to t 
=1000Nm 
at t 
=208129s 
˜ 33 orbits. 
. 
the initial conditions for the tether positions,  1 and  2 are chosen to be 
slightly asymmetrical. 

. 
the tether mass, Mtether, is the discretized mass of a section of the tether, the 

total tether mass in this case is given by nMtether =100kg. 

5Fordetails ofground testing and analysis, see[Cartmell et al.,2003]. 

117 


An asymmetryin the initial tether angles was chosen to counter numerical integrationproblems 
–thenumerical solverhasgreatdifficultiesinintegratingtheequations 
of motion when the tether arms are exactly 180. apart. The small asymmetry is 
acceptable, as the tether arms will never be exactly opposite. 

The solution to the equations of motion for the above initial conditions are given 
in Figures 5.2 to 5.4. 

5.5.1 Discussion of results for deployment 
As can be seen in Figures 5.2a, the radius vector R 
is constant – this is due to the 
assumption intheexpressionforpotential energy thatthemassesactatdistance R 
from the inertial origin. 

ThemotortorqueshowninFigures5.2bisincreased linearlytoamaximumtorque 
to avoid any large start-up transients. This is key to ensure a smooth deployment 
phase. A slow increase in the torque results in a gradual increase in the tether 
rotational velocity, and a gradual increase in the stress in the tether. 

The length of the payload 1 tether smoothly reels out from the hub, as shown 
in Figures 5.2c, first slowly, then speeds up to a moderate and predictable linear 
increase due to the tether braking6 . 

Ifthetorquewassuddenly increasedfromzerotomaximum,the initialfewminutes 
ofdeploymentwouldbeuncontrolled,and thisis likelytocausesignificantproblems 
in several areas – including tether asymmetry and nonlinear flexing in the tethers. 
This would invalidate the previous assumptions, perhaps leading to payload/stator 
interaction, which is to be avoided at all costs. 

The linear scaling in torque undoubtedly delays the overall time taken to deploy 
the tether, compared to a flat maximum torque profile, and could be optimised 

6Recall that the tether braking force is proportional to the square of the tether velocity: 
L.
2

F 
.-Ldamp 


118 


further. This isaconservativeapproach toensurethesafedeploymentof thetether, 
and inevitably leads to a longer deployment time than is absolutely necessary. 

The torque drives the tether rotational speed, as shown in Figures 5.2d: it first 
increases, then trends back to an asymptote at around 1.35rad/s. 

The profile of angular velocity varying with length is shown in Figure 5.3b. This 
shows that the rate of increase of  . 1 is within acceptable limits and will not place 
onerous demands on the strength of the tether. 

The potential 
V 
the payload may achieve is defined as the dependence between 
therotationandlength,
V 
=  . 
1L1,and isshown inFigure5.3a. The
V 
increases 
approximately linearly,incontrasttothenonlinearbehaviouroftheangularrotation 
and length generalised coordinates. At the end of the numerical integration, 
V 
˜ 
1000m/s. 

One issue with the MMET system is that there is a possibility of mutual interference 
with the tethers of the payload a stator arms. As Figures 5.3c and 5.3d 
show, in this case, the difference between the angle of the tethers and the lengths 
of the tethers are small. The angular difference is nominally zero after a start-up 
transient, with an initial maximum of ˜ 0.5., which is acceptable for the spin-up 
criteria7 . The difference in lengths of the tethers are minimal – this is likely a result 
of rounding errors encountered using a numerical solver. 

Analysing the tension and stress in the tether shows that the tether is intact and 
withinmaterialbounds. Thetethertension,shown inFigure5.4a,ataround100kN 
is high, but within acceptable limits. The stress, shown in Figure 5.4b, assumes a 
CSA of 64mm2, equivalent to a single circular section8 of diameter 9mm. 

The stress level in the tether is below failure point. The tensile strength of Zylon 

7This may be unacceptable for the final ‘launch phase’ of the payloads where the required 
accuracy of the tether angle is likely to be more stringent. 

8This is a simplifying assumption – a tether launch satellite would be likely to employ a combination 
of a tapered tether to reduce the mass along the span of the tether and a multi-strand 
tether to mitigate against single-strand failure. 

119 


fibre is5.9GP 
a;with a safetyfactor of1.5, the maximum acceptable tensile strength 
is3.9GP 
a.Themaximumcalculated stress inthetetheris1.6GP 
a, whichprovides 
an additional level of redundancy in the tether. The total safety factor is approximately 
15.
.69 =3.7. 

The stress in the tether has been designed to be linear, avoiding step-increases in 
thestressand allowing the load todistributeevenlyalong the fibre inaprogressive 
way. 

The tether tension is calculated as the sum of the tension due to the tether mass 
and the payload: 

1 22 2 

T 
= 


AL1  . 1 + L1 M1  . 1 (5.21) 
2 
where A 
isthe cross-sectional area(CSA) of thetether, and . 
is the tether density. 
The tether stress is defined conventionally as: 

T 


s 
= (5.22) 

A 


5.6 Refining the equations of motion 
A refinement of the tether mathematical model is proposed. Whereas the previous 
model describes a discretized tether of constant mass, Mtether, this can be more 
accurately modelled as a mass proportional to the tether length: 

AL 


Mtether = (5.23) 

n 


where . 
is the tether density, and Ln 
is the effective distance between mass points. 

As shown previously in Figure 5.1, the tether mass model allows the dynamics of 
a varying tether mass to be investigated. This is more accurate, better reflecting 
the realities of the mass distribution when the tether is deployed. 

120 


The previous set of equations modelled the mass points as fixed fractions of the 
tether which moved proportional to each other and did not vary in mass. This is 
unsatisfactory because the entire tether mass would be distributed over the length 
of the tether, whether the tether length is a few metres or at full deployment. 
Obviously, this is not representative of the tether mass therefore a new model is 
proposed to better describe the tether mass in Equation 5.23. This refines the 
tether mass model to increase the tether mass in proportion to the length of the 
tether, giving a more representative model of the tether mass. 

Aswith theprevious investigation intothedeployment of thetether,theequations 
of motionaresolvedforthefollowinginitial conditionsforatimeperiod9 of69376s: 

M1 = M2 =10kg, 
Mtether = AL/n, 
Mfacility =500kg, 
. 
=1570kg/m3,A 
=(0.008)2 =0.000064m2,n 
=10 
R.
[0] =0,.
[0] =0.00113905rad/s, 
 . 
1[0] =0.1rad/s, 
 . 
2[0] =0.1rad/s, 
L. 1[0] =0,L. 2[0] =0,R[0] =7.378* 106 m, 
[0] =0, 1[0] =0.01., 2[0] =180.1. , 
L1[0] =100m, 
L2[0] =100m, 
max =1000Nm, 
Ldamp =5* 108 kg/m 


Toavoidtheproblemofinitiallyover-accelerating thepayloads,anonlineartorque 
is applied by the tether motor, as shown in Figure 5.5b. This effectively limits 
the initial spin-up torque, limiting the angular acceleration and providing a stable 
basis for controlled deployment. The maximum value of angular acceleration is 
0.025rad/s2, and the maximum angular speed is 10.2rad/s. Figure 5.5d shows a 
time-history of  .
with a moderate angular acceleration in the first few seconds of 
spin-up,followedby aprogressive and controlleddeployment ofthetether thereafter. 

The torque is controlled in such a way to avoid an initial spike of acceleration, to 
progress rapidly to a high-torque plateaux and furthermore, to provide a smooth 

9This corresponds to exactly 11 orbits. 

121 



transition between the two. 

-7500 

t 
= max 
· exp +50 (5.24) 

t 
+1 

where the maximum torque provided by the motor, max 
˜ 1000Nm 
and t 
is the 
elapsed time in seconds. 

A significant advantage of the motor torque control is that the tether is deployed 
linearly,with theintentionoflimiting thejerk oraccelerationtothetether,asshown 
in Figure 5.5c. The linear deployment is very successful in limiting the tension and 
stress in the tether, see Figure 5.7a and Figure 5.7b respectively. 

The expression for the tether tension at the facility (i.e. the point of greatest 
tension) has changed compared to the previous tether mass model described in 
Equation 5.21. The tension is not calculated with respect to the centre of mass of 
the tether in the centre of the tether, it is instead assumed that the tether mass acts 
in its entirety at the root of the tether. The tether tension in the discretized tether 
is calculated as: 

T 
= AL12  . 
12 
+ L1 M1  . 
12 
(5.25) 

where A 
is the cross-sectional area of the tether, and . 
is the tether density. 

When examining stresses in the tether, the maximum stress levels for the exponential 
torque case appears higher at 2.1GP 
a, compared to 1.3GP 
a 
for the linear 
torquemodel.This isduetothediffering tension(and therefore,stress) expressions 
in the two cases, and can not be directly compared. 

However, if the stress is qualitatively examined, it is interesting to compare the 
stress profiles. The time at which the maximum stress is applied is very different; 
with the linear torque case, the stress is maximum at the end of the spin-upjust 
before payload release; with the exponential torque case, the stress quickly rises to 
maximumearly inthespin-up,and iseased slightlybeforepayload releaseachieving 

122 



a stress before release of 1.75GP 
a. The maximum allowable stress, for comparison, 
is 3.9GP 
a 


The time taken to spin the payload to release velocity is significantly less in the 
exponential torque case – the time taken to spin-up is one third of the linear torque 
case. This is partly due to the difference in mass modelling, and partly due to the 
torque profile used. This represents a marked improvement, and will undoubtedly 
enhance the economic strengths of the MMET as a renewable launch platform. 

In terms of tether synchronisation, Figures 5.6c reveals that the tethers are well 
separated by approximately 180., and furthermore, they tend to settle at a point 
diametrically opposed to each other. This is an important result, especially when 
the tethers are initially separated by a few fractions of degrees as would occur in a 
real spin-up scenario. The solution to the equations of motion were based on the 
initial conditions  1 =0.01. and  2 =180.1. . 

Likewise, inFigure5.6d,thedifferenceintetherlengthsareminimal.Thismaynot 
be true in the real world, as material properties and coefficients of friction are very 
difficult to exactly match between the two tether arms. Clearly, the asymmetrical 
spin-up scenario requires further study. 

As it is clear that deploying the tethers with a MMET system is achievable for a 
modest payload, the recovery of the tether model shall be examined. 

5.7 Recovery of tethers 
OneofthestrengthsoftheMMETis in itsroleasareusable launcherofpayloads. In 
order to re-use the tethers, they must be de-spun and then recovered to the facility. 

Thehardwarerequired toperformthede-spinisalargebraking system –thedisc 
brake system on a car axle is similar in operation to what may be required. There 
are important differences, however, as the disc brake must be cooled. On Earth’s 

123 



roads,thecoolingisprovidedby conductiveheattransferof air; onorbit,conduction 
and radiation are necessary to channel the excess heat away from the brake to the 
vacuum of space. 

The required braking energy is significant – approximately equal to the energy 
supplied toacceleratethetethertoreleasevelocity. Thepowersuppliedby thesolar 
cells to the motor will be comparable to the power required to brake the assembly, 
assuming equal time spent accelerating and decelerating the payloads. Given that 
thesolarcellsarecurrently inthe kW 
range, thebraking system andheatdissipation 
systems should be sized accordingly and should not burdensome to source. 

5.7.1 Recovery profile 
Recovery requires a different approach to deployment – the two are conceptually 
opposite, but can not be treated as mirror images when analysing the technicalities 
as there are important differences between the two cases. 

Deployment simply requires a spinning motor and a braking system, the tether 
arms held rigid by the rotation. 

When recovering the arms, there are requirements that must be fulfilled: 

. 
the tethers must be slowed gradually to avoid backlash and tether tangling 

. 
there must be a minimum spin rate to keep the tether rigid 

. 
the tethers must be fully recovered to enable a subsequent deployment 

The initial slow-down phase is similar to the spin-up phase – energy is dissipated 
during the slow-down to minimise tension in the tether, making recovery easier. The 
motorrequiredtoreelthetether inforstoragemay require lesspower(doing less 
work against the tether tension) and therefore will be lighter and cheaper. 

124 



5.7.2 The equations of motion for recovery 
The equations of motion are the same as thedeployment equations of motionfor the 
orbital generalised coordinates, R 
and , and the spin generalised coordinates,  1 
and  2.The lengthgeneralised coordinates, L1 and L2 shown inEquations5.26 and 
5.27,differ inthefactthatthey arenotbraked tostopfastdeployment,butthey are 
retracted undermotorpower. Similarly,the initial conditionsand non-conservative 
forcing terms10 have been altered to reflect the differences between decelerating and 
accelerating the tether arms. 

The aim of the new length equations of motion is to provide a consistent length 
.

recovery rate for the tethers: L 
= -v. This models a constant speed motor reeling 
in the tethers. A small acceleration term, L¨
, isadded tomakeboth length equations 
into two ODEs, such that the equations of motion may be solved numerically. 

¨ 

0.000001 L1 + L. 1 = -Lrec 
(5.26) 
¨ 

0.000001 L2 + L. 2 = -Lrec 
(5.27) 

The non-conservative forces in the right hand side of Largange’s equations are: 

.. 


0 

.. 


.. 


..

0 

.. 


.. 


.. 


. 
-  damp 
 . 1 
. 


..

Q 
= 
. 
(5.28) 

.. 


.- .

 damp 
 . 
2 

.. 


.. 


.. 


. 
-(Lrec) 
. 


.. 


-(Lrec) 

Thenon-conservativeforcesforthe length equationsof motionprovideaconstant 
reel-in speed, modelled by the rate Lrec. Similarly, the spin generalised coordinates 

The right hand side terms of Lagrange’s equations, shown in Equation 5.28. 

125 


dictate a linear braking proportional to the rotational speed of the tethers. 

5.7.3 Solving the equations of motion for recovery 
The recovery stage necessitates a two-stage approach to recovering the tether. The 
first stagerecoversthetethersclose intothefacility,at whichpoint aslowerrecovery 
stage is used to safely guide the remaining length of tether into the facility. 

The tether arms must be kept rigid11, and this requires a minimum rotational 
velocity of thetethersystem. Thetether isdecelerated toasmallrotational velocity 
in the first phase with a constant braking force. Once the tether arm is rotating 
slowly, close to the facility, a much lower braking force and marginally lower tether 
retractionspeed areemployed toensurethesaferecovery ofthe last sectionoftether. 

The initial conditions for the first recovery phase, numerically solved for a time 
period of 50000s, are: 

M1 = M2 =10kg, 
Mtether = AL/n, 
Mfacility =500kg, 
. 
=1570kg/m3,A 
=(0.008)2 =0.000064m2,n 
=10, 
R.
[0] =0,.
[0] =0.00113905rad/s, 
 . 
1[0] =1.0rad/s, 
 . 
2[0] =1.0rad/s, 
L. 1[0] =0,L. 2[0] =0,R[0] =7.378* 106 m, 
[0] =0, 1[0] =0.01., 2[0] =180.1. , 
L1[0] =1000m, 
L2[0] =1000m, 
Torquemax 
=0Nm, 
Lrec =0.015kg/m, 
 damp =1500Ns 


Again, an asymmetry in the tether angles was chosen to reflect the unknowns that 
affect the payload release dynamics. Additionally, the numerical solver has great 
difficulties in integrating the equations of motion when the tether arms are exactly 
180. apart. 

The second phase of recovery has the initial conditions similar to the first phase, 
and is numerically solved for a time period of 25000s. Initial conditions that differ 

11This is an explicit assumption made when assembling the equations of motion. 

126 


are listed below: 

 . 1[0] =0.026707rad/s, 
 . 2[0] =0.026707rad/s, 
L. 1[0] =0,L. 2[0] =0, 
L1[0] =250m, 
L2[0] =250m, 
Lrec =0.010kg/m, 
 damp =100Ns 


The goal for recovery of the tethers are to provide a stable and controlled reel-in, 
avoiding scenariosthat would lead toabreak of thetethersuch astetherasymmetry 
or uncontrolled resonance in the tether. 

Foracircularorbitand a lineartetherrecoveryprofile,thepayload tethersremain 
separated by approximately 180. . The maximum range of tether movement from 
thesymmetriccentre lineisapproximately0.43.,asshown inFigures5.9cand5.12c. 
Furthermore, there are no significant deviations or other areas of concern in either 
the spin angle or the tension or stress profiles. 

Thetensionandstress inthetetheraregradually lowered,whilekeeping thetether 
taut, as seen in Figures 5.10a, 5.13a, 5.10b and 5.13b. This will not guarantee a 
successful recovery, but will reduce the risk of a transient phenomenon that may 
cause the tether to momentarily rise above the ultimate strength of the material, 
thus breaking the tether. 

The rotational speed of the arms initially increase due to conservation of momentum,
asthe length(andtherefore inertia) ofthetethersarereduced. Angular 
momentum isconserved,and asFigure5.9b clearlyshows,thetetherpayloadsrotate 
at a faster rate while reeled in. This does not lead to an increase in the tension or 

V 
due to the careful choice of the rotationalbraking system(andbraking constant 
 damp)and the reel-in speed(Lrec). 

The second phase of tether recovery is shown in Figures 5.11 to 5.13. This occurs 
with 250m 
of tether still outside the facility, i.e. one quarter remaining to recover. 
The tether arms are rotating slowly enough to reduce the tension in the tether to 

127 



minimal levels, while keeping the tether rigid enough to ensure the equations of 
motion remain valid in describing the motion. 

The tether arms are decelerated at a lower rate than the previous recovery phase, 
with  damp =100Ns. The tether recovery rate is slowed slightly to L.
=0.1m/s. 

Similartothe first recoveryphase,theremaining tether isslowlyretracted intothe 
facility with no adverse dynamic effects. The angle between the tethers remains at 
approximately 180. ±0.5. . However, at such a low tension, it is unknown whether 
the tether can remain rigid without coiling or exhibiting other nonlinear behaviour. 
An in-depth analysisshouldbeperformed toascertainthetether’sbehaviour insuch 
an environment, as a study that involves non-rigid tether behaviour is beyond the 
scope of this thesis. 

Thetimetakentodeploythenrecoverthetetheris69376 +(50000 +25000) = 
144376s, ˜ 40 hours, or a little under 22 orbits12, not taking into account the time 
taken to release the payloads and the time to stabilise the tethers between the two 
recovery stages. This short timespan allows a small swarm of 18 nano-satellites to 
be launched from the MMET within a month. 

As one of the key strengths of the MMET is the re-usability of the platform, this 
is an encouraging finding. 

5.8 
Deployment and recovery on an elliptical orbit 
The MMET describing an elliptical orbit may interact with the tether while deploying 
or retracting the tethers. In order to study this, a sensitivity analysis is carried 
out to ascertain if this hypothesis is correct. 

The deployment and recovery of the tether are carried out under identical initial 

12 


One orbit takes approx. 6307s 
at an altitude of 1000 km. 

128 


conditions to the equations describing the discretized mass model with the exponential 
tether tension model. 

The non-dimensional ellipse eccentricity parameter, e 
=0.25, is supplied to the 
initial conditions of Lagrange’s equations of motion through the orbital angle initial 
condition: . =0.00111382rad/s. 

This is calculated using the semi-latus rectum and orbital parameters, as in [Szebehely 
and Mark, 1998], to calculate the orbital angular velocity thus: 

The semi-latus rectum(SLR) is calculated: 

SLR 
= R(1+e 
cos()) (5.29) 

where e 
is the eccentricity. 

This isusedtocalculatetheorbital angularvelocity as in[Ziegler,2003]: 

v 2 

µ 
(1+e 
cos)

.= 3 (5.30) 

SLR 


While deploying the tether, the orbital parameter . 
does not adversely affect the 
dynamics of the tether deployment. As shown in Figure 5.14b, the key deployment 
metrics – the smooth deployment of the entire tether length and a progressive increase 
in stress over time – are realised. When the parameters for the circular and 
elliptical are compared, the parameters share much in common. When comparing 
the tether length with angular velocity, for example, Figures 5.14b and 5.6b are 
almost identical. 

Similarly, when the recovery of the tether undergoing a elliptical orbit is analysed, 
the solutions to the equations of motion present a similar result. The length and 
angular velocity are very similarly matched over the two cases, as is shown when 
comparing Figures 5.15b and 5.9b. 

129 

2 

5.9 Centre of mass movement 
Additional stresses may be exerted on the tether by centre of mass movement. This 
movementcausestheroot of thetetherto libratealong theorbitalpath,subjecting 
the tether to additional loading. 

The movement of the centre of mass is shown in Figure 5.16 for the deployment 
cases and Figure 5.17 for the recovery case. The movement of the centre of mass 
with respect to the facility mass is given by Equation 5.4. 

The maximum movement of the CoM, for the deployment cases, are 0.07m 
and 
0.0017m 
– the latter is the discretized mass model using exponential torque. The 
effort expended to model the tether mass in more detail has stripped back a layer 
of conservatism, and shown that the centre of mass movement is minimal. This 
reinforcestheearlier findingsof therotationandlengthgeneralised coordinates; the 
symmetrical angleof thetethersand theequal lengthsof thetethersboth contribute 
to a very symmetrical system. 

Themaximummovement oftheCoMfortherecovery case isanorderof magnitude 
greater, at 0.7m 
for the first phase and 0.08m 
for the second phase. This is due to 
the starting assumption of asymmetrical tether angles:  1[0] =0.01. and  2[0] = 
180.1. . 

Whereas the difference in the tether angle in Figure 5.6c at full deployment tends 
to zero, the initial conditions to recover the tether are asymmetrical. This is an 
allowanceforthepayload release,whichisanunknownquantityinMMETdynamics. 
Therefore, an initial asymmetry of 0.1. and 0.01. was chosen to investigate the 
recovery dynamics. To remedy this unknown, a separation study is recommended, 
to fully investigate the effects that a symmetrical (or asymmetrical) dual release 
have on the tethers, however this is outside the scope of this document. 

Onfurther examination,thegraphs of the magnitude ofposition and acceleration 
were plotted against time, and are show in Figures 5.18a and 5.18b. 

130 


The maximum acceleration, due to the moving CoM, encountered in the recovery 
phase is approximately 0.8m/s2, therefore the force exerted by the moving centre 
of mass,givenatotal systemmassof10+10+100+100+500 =720kg 
is: 

F 
= ma 
=0.8* 720 =576N 
(5.31) 

Giventhis isseveral ordersof magnitude lessthanthetethertensionof110, 
000N 
inFigure5.10a, itdoesnotpresent asignificant risk tothetether. Indeed, itexplains 
the slight high frequency tension superimposed on the overall smooth downward 
trend shown in that figure. 

Onpayload release,the likelihood of asmalltetherasymmetry may besignificant. 
Removing thisasymmetry, insofaraspossible,couldbeachievedby repeated cycling 
of short recovery anddeploymentphases, oranactivecontrol systemonthepayload 
release mechanism. 

5.10 Conclusions 
Deploying and recovering the tethers on the MMET system is not a trivial matter; 
however, they can be achieved in a well controlled and structured way. 

A method for deployment and recovery in the orbital plane have been outlined, 
givingconfidencethattheMMETiscapableand suitableforuseasareusable launch 
platform. 

131 



5.11 Figures 
R’ 


-10


1.5·10


-10


1·10


-11


5·10


-11


-5·10


-10


-1·10


-10


-1.5·10



t 


86400 172800 



(c) Tether 1 lengthL1 
(m)vs. time t 
(s) (d) Tether 1rotational velocity  . 1 
(rad/s)vs. 
time t 
(s) 
Figure 5.2: Initial MMET Spin-up 

Motor Torque 


1000 
800 
600 



R 


66667 77


2·104·106·108·101·101.2·101.4·10


400 
200 
t



86400 172800 


(a) Phase plot: radius velocity .R 
(m/s) vs. (b) Applied motor torque t 
(Nm) vs. time 
radius of CoM R 
(m) t 
(s) 
L1 D@.1,t. 



t 



86400 172800 
0.5 
1.5 
2 
132 



DeltaV L1 


86400 172800 
200 
400 
600 
800 
1000 
1200 
800 


600 


400 


t 


.1' 

0.8 


1.2 1.4 1.6 1.8 2 
(a) Payload 
V 
(m/s)vs. time t 
(s) (b) Angular velocity  . 1 
(rad/s) vs. length 
L1 
(m) 
.1-.2 

L1-L2 

200 
50000 
t 


0.0075 


0.005 


-5·10


0.0025 


-0.00001 


t 


100000 150000 200000 


-0.000015 


-0.0025 


-0.00002 


-0.005 


-0.0075 


-0.000025 


50000 100000 150000 200000 
-6 
(c) Difference in tether angles 1 
- 2 
-p 
(rad) (d) Difference in tether lengths L1 
-L2 
(m) 
vs. time t 
(s) vs. time t 
(s) 
Figure 5.3: Initial MMET Spin-up 

Tether 1 Stress HGPAL

TetherTension N 


86400 172800 
Tether 1 Intact 


100000 


9


1.5·10


80000 


9


1.25·10


9


60000 


1·10


8


7.5·10


40000 


8


5·10


20000 


8


2.5·10


t 


t 


86400 172800 



(a) Tether tensionT 
(N)vs. time t 
(s) (b) Tether stresss 
(N/m2)vs. time t 
(s) 
Figure 5.4: Initial MMET Spin-up 

133 


R’ 


-11


3·10


-11


2·10


-11


1·10


-11


-1·10


-11


-2·10


-11


-3·10


-11


-4·10


Torque 
R 
66667 77


2·104·106·108·101·101.2·101.4·10


800 
600 
400 
200 



t 


10000 20000 30000 40000 50000 60000 70000 


(a) Phase plot: radius velocity R.
(m/s) vs. (b) Applied motor torque t 
(Nm) vs. time 
radius of CoM R 
(m) t 
(s) 
L1 D@.1,t. 


1000 


800 


600 


400 


200 



4 


2 



t 



t


21600 43200 64800 


21600 43200 64800 



10 


8 


6 



(c) Tether 1 lengthL1 
(m)vs. time t 
(s) (d) Tether 1rotational velocity  . 1 
(rad/s)vs. 
time t 
(s) 
Figure 5.5: Initial MMET Spin-up, discretized tether mass model 

134 



DeltaV 


L1 



1000 



800


800 


600


600 


400


400 


200


200 


.1' 

t 


2 4 6 810


21600 43200 64800 


(a) Payload 
V 
(m/s)vs. time t 
(s) (b) Angular velocity  . 1 
(rad/s) vs. length 
L1 
(m) 
.1-.2 

0.008 


0.006 


0.004 


0.002 


-0.002 


-0.004 


L1-L2 

t 


10000 20000 30000 40000 50000 60000 70000 


-0.00002 


-0.00004 



t 


10000 20000 30000 40000 50000 60000 70000 -0.00006 


-0.00008 



(c) Difference in tether angles 1 
- 2 
-p 
(rad) (d) Difference in tether lengths L1 
-L2 
(m) 
vs. time t 
(s) vs. time t 
(s) 
Figure 5.6: Initial MMET Spin-up, discretized tether mass model 

135 



Tether 1 Stress HGPAL

TetherTension N 


Tether 1 Intact 



9


120000 


2·10
100000 


9


1.5·10


80000 
60000 


9


1·10


40000 


8


5·10


20000 
t 


t


21600 43200 64800 


21600 43200 64800 



(a) Tether tensionT 
(N)vs. time t 
(s) (b) Tether stresss 
(N/m2)vs. time t 
(s) 
Figure 5.7: Initial MMET Spin-up, discretized tether mass model 

D@.1,t. 


L1 
1000 
1 


900 


800 


0.8 
700 
0.6 


600 



0.4 



500 


400 


0.2 
t 
43200 



t 



21600 43200 


21600 
(a) Tether 1 lengthL1 
(m)vs. time t 
(s) (b) Tether 1rotational velocity  . 1 
(rad/s)vs. 
time t 
(s) 
Figure 5.8: MMET reel in, first phase 

136 



DeltaV 


L1 


21600 43200 
200 
400 
600 
800 
1000 
1000 


900 


800 


700 



.1' 

1.08 1.09 
1.11 1.12 1.13 1.14 
t 


(a) Payload 
V 
(m/s)vs. time t 
(s) (b) Angular velocity  . 1 
(rad/s) vs. length 
L1 
(m) 
L1-L2 

.1-.2 

10000 20000 30000 40000 50000 
-0.0075 
-0.005 
-0.0025 
0.0025 
0.005 
0.0075 
1 


0.5 


t 


t 


10000 


20000 


30000 


40000 


50000 


-0.5 


-1 


(c) Difference in tether angles 1 
- 2 
-p 
(rad) (d) Difference in tether lengths L1 
-L2 
(m) 
vs. time t 
(s) vs. time t 
(s) 
Figure 5.9: MMET reel in, first phase 

137 



21600 43200 
t 
20000 
40000 
60000 
80000 
100000 
TetherTension N 
(a) Tether tension T (N) vs. time t (s) 
21600 43200 
t 
2.5·108 
5·108 
7.5·108 
1·109 
1.25·109 
1.5·109 
1.75·109 
Tether 1 Stress HGPAL 
Tether 1 Intact 
(b) Tether stress  (N/m2) vs. time t (s) 
Figure 5.10: MMET reel in, first phase 
3600 7200 10800 14400 18000 21600 
t 
50 
100 
150 
200 
250
L1
(a) Tether 1 length L1 (m) vs. time t (s) 
3600 7200 10800 14400 18000 21600 
t 
0.005 
0.01 
0.015 
0.02 
0.025 
0.03 
D@.1,tD 
(b) Tether 1 rotational velocity  .1 (rad) vs. 
time t (s) 
Figure 5.11: MMET reel in, second phase 
138

DeltaV 


L1 



t 


3600 7200 10800 14400 18000 21600 


(a) Payload 
V 
(m/s)vs. time t 
(s) (b) Angular velocity  . 1 
(rad/s) vs. length 
L1 
(m) 
L1-L2 

.1-.2 
1 


0.0075 


0.005 


0.5 


0.0025 


t 


t 


25000 


5000 


10000 


15000 


20000 


25000 


-0.0025 


-0.5


-0.005 


-0.0075 


-1 


(c) Difference in tether angles 1 
- 2 
-p 
(rad) (d) Difference in tether lengths L1 
-L2 
(m) 
vs. time t 
(s) vs. time t 
(s) 
Figure 5.12: MMET reel in, second phase 


0.005 0.01 0.015 0.02 0.025 0.03 
.1' 
50 
100 
150 
200 
250 
5 
4 
3 
2 
1 


5000 10000 15000 20000 
139 



TetherTension N 


6 


5 


4 


3 


2 


1 



t 



3600 7200 10800 14400 18000 21600 


(a) Tether tensionT 
(N)vs. time t 
(s) 
Tether 1 Stress HGPA. 
Tether 1 Intact 

100000 
80000 


60000 


40000 


20000 



t 


3600 7200 10800 14400 18000 21600 


(b) Tether stresss 
(N/m2)vs. time t 
(s) 
Figure 5.13: MMET reel in, second phase 

R 


7


1.2·10


7


1.1·10


7


1·10


6


9·10


6.28319 12.5664 18.8496 25.1327 31.4159 37.6991 43.9823 



(a) 
L1 
1000 


800 


600 


400 


200 


T 
.1' 

2 4 6 810 



Orbital radius R 
(m) vs. orbital angle (b) Angular velocity  . 1 
(rad/s) vs. length 

. 
(rad) L1 
(m) 

Figure 5.14: MMET deployment with elliptical orbit 

140 


6.283185301721.75965683674076161894.3854991575259952321955.31837257944132032818.741185394256950385897932385 
T 
9·106 
1·107 
1.1·107 
1.2·107
R 
(a) Orbital radius R(m) vs. orbital angle 
 (rad) 
1.08 1.09 1.11 1.12 1.13 1.14 
.1' 
700 
800 
900 
1000
L1 
(b) Angular velocity  .1 (rad/s) vs. length 
L1 (m) 
Figure 5.15: MMET recovery with elliptical orbit 
-0.1 -0.05 0.05 0.1 
CoM_x 
-0.1 
-0.05 
0.05 
0.1 
CoM_y 
(a) Centre of mass position, linear mass model 
-0.0015 -0.001 -0.0005 0.0005 0.001 
CoM_x 
-0.0015 
-0.001 
-0.0005 
0.0005 
0.001 
0.0015 
CoM_y 
(b) Centre of mass position, discretized mass 
model 
Figure 5.16: MMET deployment, centre of mass movements 
141

-1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1 
CoM_x 
-1 
-0.75 
-0.5 
-0.25 
0.25 
0.5 
0.75
1 
CoM_y 
(a) Centre of mass position, discretized mass 
model, first stage 
-0.1 -0.05 0.05 0.1 
CoM_x 
-0.1 
-0.05 
0.05 
0.1 
CoM_y 
(b) Centre of mass position, discretized mass 
model, second phase 
Figure 5.17: MMET recovery, centre of mass movements 
10000 20000 30000 40000 50000
t 
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 
AbsHCoML 
(a) Absolute position of CoM, discretized 
mass model, first phase 
10000 20000 30000 40000 50000
t 
0.2 
0.4 
0.6 
0.8 
AbsHCoM_aL 
(b) Absolute acceleration of CoM, discretized 
mass model, first phase 
Figure 5.18: MMET recovery, position and acceleration of centre of mass 
142

143 


5.11.1 Linear kinetic energy 
The linear kinetic energy of the MMET with length deployment and ten discrete tether masses, expanded from the analytical version 
in Equation 5.12, is as follows: 

2 
Ttrans = Mtether(11M1 (M2 + Mfacility)+145M1 Mtether +5Mtether (7(M2 + Mfacility)+85Mtether))L. 1 +(11M2 (M1 + Mfacility) 


2)L. 22 2 ..2 ..R.
2

+5(7M1 +29M2 +7Mfacility)Mtether +425Mtether +11M1 R2 +22M1 M2 R2 +11M2 R2 +22M1 Mfacility R2 +22M2 Mfacility 
2 ....2 .

+11Mfacility R2 +440M1 Mtether R2 +440M2 Mtether R2 +440Mfacility Mtether R2 +4400Mtether R2 +11M12 R2 .2 +22M1 M2 R2 .2 
+11M22 R2 .2 +22M1 Mfacility R2 .2 +22M2 Mfacility R2 .
2 +11Mfacility 
2 R2 .2 +440M1 Mtether R2 .
2 +440M2 Mtether R2 .2 
+440Mfacility Mtether R2 .
2 +4400Mtether 
2 R2 .2 +22(M1 +5Mtether)(M2 +5Mtether)L1 sin( 1 - 2)L. 2  . 
1 
2222 22
2222 22

+11M1 M2 L1  . 1 +11M1 Mfacility L1  . 1 +145M1 Mtether L1  . 1 +35M2 Mtether L1  . 1 +35Mfacility Mtether L1  . 1 +425Mtether 
2 L1  . 1 

-22(M1 +5Mtether)(M2 +5Mtether)cos( 1 - 2)L1 L2  . 1  . 2 +(11M2 (M1 + Mfacility)+5(7M1 +29M2 +7Mfacility)Mtether 
2

2

+425Mtether 
2)L2  . 2 -22(M1+5Mtether)(M2+5Mtether)L. 1 (cos( 1- 2)L. 2+L2 sin( 1- 2) . 2)22(M1+M2+Mfacility+20Mtether) 

(5.32) 

Chapter 6 

Dynamics of space-webs 

This chapter was funded as part of the ESA Ariadna study into space-webs; the 
final report waspresented as[McKenzie et al.,2006].The authorwishestoacknowledge 
the hard work and contributions to this report from M.P Cartmell and M. 
Vasile of the University of Glasgow and D. Izzo of the ESA ACT. 

6.1 Introduction 
As was outlined in Section 2.3, a generic satellite that may be constructed or re-
configured asneeded isauseful concept. Studieshavepreviouslyshownthat robots 
may be deployed in this way to reconfigure the net structure [Kaya et al., 2004a], 
[Nakano et al., 2005]. 

This chapter will consider the fundamental conceptual design of an appropriate 
and generic thin membrane, its orbital mechanics and control. 

A model of a net in orbit is presented here, with robots moving along the surface 
of the net, simulating reconfiguration of the system. Useful guidelines to the initial 
conditions of the net and components arefound andgive a startingpointforfurther 

144 



studies into the field. 

This new class of structures with robots moving over the surface will be defined 
as ‘Space-webs’, with the robots correspondingly named ‘Spiderbots’. A concept 
illustration of the idea is shown in Figure 6.1. 


Figure 6.1: Concept rendering of the space-web with spiderbots on the surface. 

Space-webs have one definite advantage over single use spacecraft: the ability to 
reconfiguretheweb tosuitthemissionorenvironment. Duediligencemusttherefore 
be observed to ensure that the act of reconfiguring the web does not endanger the 
web or satellite. Investigating the movement of the robots while moving over the 
web is essential: the movement of their mass and momentum may significantly affect 

145 



the web dynamics. 

The stability of the system is investigated while the robots crawl along the web 
surface in two pre-defined motions: 

. 
Three robots simultaneously moving along the outer catenaries of the web 

. 
Three robots simultaneously moving from the centre to the outside of the web 

6.2 Modelling methodology 
This study will aim to develop an analytical model of the fully deployed space-web 
system that may be numerically integrated. 

When constructing a numerical simulation of an existent system, it is important to 
consider that the simulation can only be said to represent a real life scenario when 
the model has been validated and the initial conditions used in the calculation are 
representative of the simulation scenario. 

This may seem like an obvious statement, but these facts are often overlooked for 
convenience or speed. 

In modelling the space-web, it is important to make the modelling process transparent 
and verifiable. In this spirit, the steps followed to assemble the space-web 
model are outlined paying particular attention to the steps followed to produce a 
model in Mathematica and the process of validating the model against previous 
models and the real life case where possible. 

146 



6.3 Web meshing 
6.3.1 Web structure 
Togain an accurate estimate ofthe energy of the web, the massdistribution over the 
web is discretized [Ziegler, 2003]. An algorithm has been designed in Mathematica 
to take the triangle bounded by the ith 
sub-span, the (i 
+ 1)th 
sub-span and the 
facility mass and divide this into equally sized smaller triangles. 

The space-web is modelled as s 
sub-spans, clustered around the central facility 
mass. Each sub-span is considered to be rigid and massless, held rigid by the 
centripetal force of rotation around the centre of mass. An idealized point mass mi 
isplaced attheend of thesub-span,at length li, with theposition of each endpoint 
mass given by Pilocal in Equations 6.5, 6.6. In this form, this is very similar to 
thetetherspreviously modelled,thedifferences aremultiple sub-spans and massless 
tethers. 

The web is stretched between the sub-spans i 
and i 
+1 and replicated to give s 
triangular webs. Each triangular web section is idealised as an elastic plane containing 
the mass of that section, however the internal elastic forces within the web 
are not modelled at this stage. 

6.3.2 Dividing the web 
Each web section is divided into n 
equal sections(referred tohereas ‘divisions’), in 
thedirectiongivenbythe linejoining themidpointofthesub-spanendsandthe 
facility, as represented in Figures 6.2 to 6.4. 

For each of the n 
web sections; 

147 



triangles = (n 
+ 1)2 
triangles in top row = 2n 
+ 1 
row configuration = {2n 
+ 1, 
2(n 
-1)+ 1, 
2(n 
-2)+ 1, 
. 
. 
. 
, 
3, 
1} 
number of rows = n 
+ 1 
nodes = 1 
2(n 
+ 2)(n 
+ 3) 
midpoints = (n 
+ 1)2 


Figure 6.2: 0 divisions Figure 6.3: 1 division Figure 6.4: 2 divisions 

6.3.3 Web divisions 
A point mass is placed at each mid-point, therefore the total number of masses for 
the webs are s(n 
+1)2, where s 
is the number of sub-spans. That is to say, the 
number of masses in the web increases as the square of the number of divisions. 
Thiscan lead toavery largenumberof masspointstoconsiderforafine-grainweb. 

Assembling the space-web withthree sub-spans(n 
= 3) and 0 divisions gives the 
layout shown inFigure6.5,withFigure6.6showing thesame layout with1division, 
and finally Figure 6.7 shows the same layout with 2 divisions. Each red dot is a 
mass-point on the web. 

Figure 6.5: 3 sub-spans 

– 0 divs 
Figure 6.6: 3 sub-spans 

– 1 div 
Figure 6.7: 3 sub-spans 

– 2 divs 
148 



Figure 6.9: 5 sub-spans 

Figure 6.8: 4 sub-spans 

Figure 6.10: 6 sub


– 2 divs 
– 1 div 
spans – 3 divs 

The space-web layout may be expanded to analyse different web configurations. A 
square layout with1divisionpersection isshown inFigure6.8; apentagonallayout 
with 2 divisions per section is shown in Figure 6.9; and a hexagonal layout with 3 
divisions per section is shown in Figure 6.10 with total of 103 masses1 . 

6.4 Equations of motion 
6.4.1 Centre of mass modelling 
The mass of the web is likely to be unevenly distributed. Modelling this requires 
anexpressionforthecentreof mass(CoM) position, inthiscaseusing thefacility 
mass as the origin. 

For n 
masses with positions {Xi,Yi,Zi}, the position of the centre of mass about 
the facility, in terms of the local {X, 
Y, 
Z}coordinate system centred on the facility 
mass, is: 

. 
. 
n 
n 
n 
. 
. 
. 
. 
. 
MiXi 
MiYi 
MiZi 
. 
. 
. 
. 
i=1 i=1 i=1 Pfacility.CoM 
n 
, 
n 
, 
n 
(6.1) 
. 
. 
. 
. 
. 
. 
Mi 
Mi 
Mi 
. 
. 
. 
i=1 i=1 i=1 

This isused whentaking theposition from the centre of mass to the facility mass, 
as PCoM.facility 
= -Pfacility.CoM 


1103 masses = 96 web midpoints + 6 sub-span masses + 1 central facility mass. 

149 


P_1 


CoM 
P_s-1 
P_s 
Facility 
P_2 
P_3 
Figure 6.11: Simplified space-web layout with s 
sub-spans 

6.4.2 Rotations 
Rotational matrices are used to rotate the starting vectors to their positions on the 
local and inertial planes. The local position vector in Equation 6.5 is the matrix 
product of a series of rotations. A diagram of the space-web configuration is shown 
in Figure 6.13 with the web removed for clarity. 

.. 


10 0 

.. 


.. 


..

R i,X = 0 cos( i) -sin( i) (6.2) 

.. 


.. 


0 sin( i) cos( i) 

.. 


cos(i) 0 sin(i) 

.. 


.. 


..

Ri,Y = 0 10 (6.3) 

.. 


.. 


-sin(i) 0 cos(i) 

150 



(i,j+1) 
(i+1,j) 
(i,j) 
L 
2 
L 
2 
h 
2h 
3 
3 
(i,j) 
L 
2 
L 
2 
h 
2h 
3 
3 
Figure 6.12: Midpoint location 

.. 


cos(i) -sin(i)0 

. 
. 
. 
. 
Ri,Z = . 
. 
sin(i) cos(i) 0 . 
. 
(6.4) 
. 
. 
0 0 1 
PiLocal = R i,X · Ri,Y · Ri,Z · Li 
(6.5) 

where Li 
= {0, 
0,li}T 
, aligned along the local Z-axis 

Piinertial = R,Z (P0.CoM -Pfacility.CoM +(R i,X · Ri,Y · Ri,Z · Li)) (6.6) 

Rotations may be performed using the rotation order Z 
then Y 
then X: 

The sub-span is rotated around the facility by an angle . 
with the axis of rotation 
intheX-axisonly. Thisvector isthenrotated around thefacilityby anangle a 
with 
the axis of rotation in the Y-axis only. Finally, this vector is then rotated around 
the facility by an angle f 
with the axis of rotation in the Z-axis only. 

151 


y2 
y1 
q 
y3 
localZ 
Ylocal 
Earth 
inertialY 
inertialX 
....
....









Mass3 
Mass2 
CoM 
....


..

Facility 
Radius of CoM 
Orbital Path of 
Space-Web CoM 
Mass1 
....
....









Mass3 
Mass2 
CoM 
....


..

Facility 
Radius of CoM 
Orbital Path of 
Space-Web CoM 
Mass1 
Figure 6.13: Space-web diagram with 3 sub-spans shown in inertial space. Note: 
the local axes are in the Y-Z plane. 

The angles are projected about their respective axes i.e., the Z-rotation, , is 
contained in the X 
- Y 
plane. The angles themselves can be compared to the 
standard aerospace rotations, with the sub-span direction from the facility to the 
edge taken as the tail-nose direction. In this case: . 
is the yaw angle, in the plane 
of the space-web; a 
is the pitch angle, out of the space-web plane; and f 
is the roll 
angle, the twist in the sub-span. 

Considering rotations about the . 
direction only – that is constraining the motion 
of the space-web to be in-plane – gives the following equation 

Piinertial = R,Z (P0.CoM -Pfacility.CoM +(R i,X · Li)) (6.7) 

Where P0.CoM = {R, 
0, 
0}, and representstheposition of the centre of massfrom 
Earth. 

152 


6.4.3 Position equations 
The positions of the masses in the local coordinate system must be translated and 
rotated into the inertial (Earth centred) coordinate system. The local position 
vectors are addedtotheposition vectorfromtheCoMtothefacility andtheposition 
vectorfromtheEarthtotheCoM.Theresultant vector isthenrotatedintotheEarth 
centred Inertial axis system. 

6.5 Energy modelling 
For every mass point in the system with position in the inertial frame, Pi, the 
following steps are undertaken in order to find the Lagrangian energy expression L: 

the respective velocities, Vi 
are found: 

. 


Vi 
= Pi 
(6.8) 

@t 


thekinetic energies(linear Tlin and rotational Trot)are obtained and summed: 

n 
1 

Tlin = miVi 
· Vi 
(6.9) 

2 

i=1 

 !2

n 
1  . 1 +  . 2 +  . 3 

· (6.10) 

Trot = mi 
Pilocal Pilocal 

23 

i=1 

the potential energies, U, are obtained and summed: 

n 


µmi

U 
= v (6.11) 

Pi 
· Pi

i=1 

and the Lagrangian is found: 
L 
= Trot + Tlin -U 
(6.12) 
153 


Themomentofinertiaof each masspoint onthewebiscalculated usingtheparallel 
axis theory about the central facility, then multiplied by the square of the average 
angular velocities of the three sub-spans to give the rotational kinetic energy. 

Theaverageangularvelocityofthethreesub-spans,asshown inEquation6.13,was 
chosen in preference to the actual calculated angular velocities of each mass point 
tosimplify theLagrangianenergy expressionand to lightenthecomputationalload. 
Instead of27individual angularvelocities,every masspoint ontheweb wasassumed 
tohaveonecommonangularvelocity. There isanerror inthisassumption,butthis 
will be acceptably small because the actual velocity is approximately equal to the 
average velocity. 

@. 
1 @ 1 @ 2 @ 3 

= + + (6.13) 

@t 
3 @t 
@t 
@t 


ave 


For each mass point, the Lagrangian energy expression is constructed by considering 
the total kinetic and potential energies of the system. 

d 
@T 
@T 
@U 


- += Qj 
(6.14) 

dt 
@q.j 
@qj 
@qj 


Lagrange’s equations are generated for all the generalised coordinates as specified 
in Equation 6.14. 

6.5.1 Virtual work terms 
The elastic force the web imparts on the sub-span ends are modelled as a simple 
elastic tether [Miyazaki and Iwai, 2004], obeying Hooke’s Law: F 
= Kx: 

2p 


F 
= KH 
(Pi 
-Pi+1)- (6.15) 

s 


where H[.. .] is the Heaviside function. 
The forces are then used to calculate the right hand side of Lagrange’s equations, 


154 



Qi, through consideration of the virtual work. 

@i

Qi 
= -Ksci 
; (6.16) 
qi 


where i 
isthedifferencebetweenthesub-spansdefinedby the in-plane( ) coordinates 
as: 

1 =0 (6.17) 

2 =0 (6.18) 

2p 
2p 


3 =  2 - 1 - H 2 - 1 - 

ss 


2p 
2p 


-  1 - 3 - +2p 
H 1 - 3 - +2p 
(6.19) 

ss 


2p 
2p 


4 =  3 - 2 - H 3 - 2 - 

ss 


2p 
2p 


-  2 - 1 - H 2 - 1 - (6.20) 

ss 


2p 
2p 


5 =  1 - 3 - +2p 
H 1 - 3 - +2p 


ss 


2p 
2p 


-  3 - 2 - H 3 - 2 - (6.21) 

ss 


Note: iftheanglebetweensub-spans1 and3( 1 -  3) is examined, this would 
be usually be large and negative – e.g. for an equally spaced space-web. this angle 
would be -240. instead of 120. . Therefore to counteract this, an offset of 2p 
has 
been added to Equation 6.19 to ensure that the angle is positive and similar in size 
tothe other sub-span angles( 2 - 1 and  3 - 2). 

155 


6.5.2 Simplifying Assumptions 
TosolvethefullsetofLagrange’sequationsforthisspace-web system inareasonable 
time on a desktop PC requires some simplifying assumptions to be made. 

Firstly, five generalised coordinates are chosen defining the space web rotating in 
theplane normal to the radius vector: (R, 
, 
 1, 2, 3). Thediagram inFigure6.13 
shows the layout of the space-web, where R 
is the orbital radius, . 
is the true 
anomaly,  n 
is the anglebetween the nth 
sub-span and the Xlocal 
coordinate system. 

Iftheout-of-planemotionofthespace-web weretobe included,thiswould increase 
thenumber ofgeneralised coordinatestoeight, aseach sub-span musthavetheoutof-
plane motion defined. 

If thespace-web orbitsexclusively in-plane,asimplifying assumptionmay beperformed 
in terms of the potential energy expression: 

n 


µmi

U 
= 
R 
(6.22) 
i=1 

6.5.3 Robot Position Modelling 
Robots may move across the space-web in order to reconfigure the web or other tasks 
as discussed previously. To model this as an kinetic energy based term while retaining 
the flexibility of defining the path without hard-coding this into the equations, 
the robot position vector needs to be kept in a very general form. 

Therobotpositionvector inEquation6.23 isdefined similarlytoEquation6.6: a 
vector addition of the R vector, the CoM vector and the vector defining the local 
position of the robot where 
{P 
RobotX 
, 
P 
RobotY 
, 
P 
RobotZ 
}local 
arethe local vectorpositionsof therobot inthe 

156 



X, 
Y, 
Z 
axes with the origin coincident on the facility mass. 

P 
Robotinertial 
= R,Z(P0.CoM 
-Pfacility.CoM 
+ 

(6.23) 
{P 
RobotX 
, 
P 
RobotY 
, 
P 
RobotZ}local) 

Therobot localpositionvectormaybeafunctionof time,positionof thesub-span 
masses ( 1, 2, 3), an arbitrary path function or any other smooth generalised 
function. 

The kinetic energies of the robot (both linear and rotational) are derived from 
the positions of the robot as before. Both the Lagrangian energy expression and 
Lagrange’sequationscontaintermsforthe localposition,velocity,and acceleration 
of the robots. 

Keeping the robot position functions in the most general form allows for rapid 
reconfiguration of the robot paths and may lead to, for example, robot control 
studies in the future. 

6.6 Space-web dynamics 
6.6.1 Dynamical simulation 
The space-web dynamics are heavily governed by the centre-of-mass movement of 
the system. The space-web system is very rarely symmetrical(both in reality and 
in simulations) and this asymmetry can lead to unstable and even chaotic motion 
in certain configurations of the space-web. 

Several different parameters were considered to be possible influences on the stability 
of the space-web system, including: 

. 
variation of the masses of the web 

. 
the masses of the robots, the sub-span and facility masses 

157 



. 
the position and velocity of the robots 

. 
the general angular momentum of the robots 

. 
the angular velocity of the web 

. 
and the starting configuration of the web or ‘web symmetry’ 

Lagrange’s equations for the space-web system are solved using the Mathematica 
numerical integrator NDSolve[. . .]. The package LiveGraphics3D [Kraus, 2007] is 
then used to display any animated graphics in a web-browser. 

Additionally,thearrowsusedto indicatedirection intheCoMplotsweregenerated 
with the CurvesGraphics6 package [Gorni, 2008] for Mathematica 5.1. 

6.7 Investigating the stability of the space-web 
To test the stability of the space-web with robot movement over the web, several 
different numerical experiments have been considered, each investigating the impact 
the following have on the centre of mass movement of the space-web: 

. 
web mass 

. 
robot mass 

. 
web asymmetry 

. 
robot crawl velocity 

6.7.1 Mass of the web 
The mass of the web was found to have a negative impact on the stability of the 
system. The higher the mass of the web, the more likely the triangle is to deform 
from the perfect equilateral shape, and the more likely the system is to exhibit 

158 



chaotic motion. This is likely to have been due to the increase in the ratio of the 
web mass to the other masses, causing the web mass to dominate the other mass 
terms. Figures B.1, B.2, B.3, B.4, B.5, B.6, B.7 and B.8 demonstrate this effect, 
which were simulated with Case 12 in which only the mass of the web is varied. 

The initial conditions used to numerically integrate the equations are given in 
Table C.1. 

The mass of the web reaches a limit of around 40 kg, after which, the web is no 
longer able to keep its shape in this configuration. 

Runs E01–E04 show the space-web can be stabilised while the robots move across 
the surface, the mass of the web increasing to a critical value of approximately 25% 
of the total system mass. The maximum CoM displacement is 0.85m 
for these four 
runs. In runs E05–E06, the mass of the web is increased to 44.82kg 
and 48.25kg 
respectively. This causes significant instability in the web for the 48.25kg 
web, 
showing that for certain configurations of the space-web system, the mass of the 
web is a critical parameter. The marginal increase in web mass gives a noticeable 
degradation in the behaviour of the space-web past that critical point. 

6.7.2 Robot mass 
The momentum of the robot was thought to be a significant parameter in the stability 
of the space-web, therefore the robot mass was investigated as a possible 
parameter. Runs 18-21 tested values of Mrobot3 from 1kg 
to 100kg 
and the CoM 
positions are shown in Figures B.9, B.10, B.11, B.12, B.13 and B.14. 

The initial conditions used to numerically integrate the equations are given in 
Table C.4. 

Counter-intuitively, increasing the mass of the robot caused the space-web to be


2With symmetrical robot paths. 
3From 3% to 17% of the system mass. 


159 


comemorestable!This isduetothecentripetal effect of moremassontheoutsideof 
the web causing the sub-spans to become more rigid and effectively act as a damper 
to the system. 

There is a sharp divide between the stable behaviour of the robot masses set 
at 16kg 
shown in Figure B.12, and the unstable behaviour with the masses at 
15kg 
shown in Figure B.11. This critical dividing line between the acceptable and 
unacceptable system movement is clear as it is unexpected, and raises significant 
issues for the construction of the space-web, especially with a lighter robot. 

6.7.3 Web asymmetry 
Thesinglebiggestdriverof instability onthespace-web systemwasfound tobethe 
asymmetry in the web configuration and mass distribution. 

Very small asymmetries in the initial conditions of the space-web (compared to 
a perfectly symmetrical equilateral triangle) are amplified in certain configurations 
of the space-web, and may create large instabilities in the system, especially for 
energetic configurations such as high web mass and/or large angular velocities. A 
smalldifference inorientation(ontheorderof1.)of the sub-spans maylead to large 
asymmetriesbetweenthesub-spans inarelativelyshorttime(˜ 100s). 

The cases chosen to investigate the effect of asymmetry on the web are as follows: 

. 
Case 1: Robots 1,2,3 walk along the edges of the web from sub-span to next 
sub-span – (symmetrical) 

. 
Case2:Robots2,3walkalongtheedgesof theweb whileRobot1 isstationary 
onsub-span1 –(unsym.) 

. 
Case 3: Robot 3 walks along the edges of the web while Robots 1,2 are stationary 
onsub-spans1,2 – (unsym.) 

. 
Case4:Robots1,2,3walkalong fromthecentretotheirsub-spans – (sym.) 

160 



. 
Case 5: Robot 1 spirals outwards from the centre to the edges, Robots 2,3 are 

fixed on sub-spans – (unsym.) 

. 
Case 6: Robot 1 walks along from the centre to sub-span1, Robots 2,3 remain 

in centre –(unsym.) 

The initial conditions used to numerically integrate the equations are given in 
Table C.2. 

Cases 1 and 4 are symmetrical, shown in Figures B.15 and B.18. Case 1 has a 
stable CoM position throughout the simulation, with a maximum CoM travel of 
only 0.46m. Case 4 is symmetrical, however has a maximum CoM travel of 10.96m 
and is on the limits of what is defined as acceptable movement for the CoM. In 
contrast, Cases 2, 3, 5 and 6 are asymmetrical, shown in Figures B.16, B.17, B.19, 
and B.20. The asymmetric cases show a large CoM movement of between 18m 
and 
38m, and are unstable. 

Compounding theproblemof unevenmass,the initial conditionsof thethreesubspans 
were found to influence the stability of the system. The equations could not 
be solved with . 
= {0. , 
120. , 
240.}or spacing the web sub-spans by exactly 120. , 
for reasons not known at this time. Initial conditions for . 
were implemented as a 
triplet; the three sub-span values of {0+ , 
120. , 
240. - }offer a solution to this 
problem. 

Moregenerally, it isexpected thatformostconfigurationsof thetriangularspaceweb, 
holding the three sub-spans permanently rigid at exactly 120. separation will 
be impossible as small perturbations to the web in the space environment will be 
constantly experienced. 

Adding the effect of many asymmetrical robot masses exacerbates the problem of 
uneven mass distribution. To remedy this, the space-web must be configured to 
occupy as low an energy state as the mission will allow: low web mass; light, slow 
moving robots; low angular velocity of the web. In this state, the asymmetry of the 

161 



web still exists, but it is kept to a manageable level. 

6.7.4 Robot crawl velocity 
Thefastertherobotstravelled along theweb,thegreaterthe likelihood of unstable 
behaviouroftheweb,astheCoMdisplacementplotsshow inFiguresB.21 andB.25. 
Both are simulated with Case 1 – symmetrical robot paths. 

The initial conditions used to numerically integrate the equations are given in 
Table C.3. 

When robot crawl velocity was increased from 0.1m/s 
to 10m/s, by lowering the 
time fixedfrom1000s 
to10s,themaximumCoMdisplacement increased marginally 

– 
from 0.3m 
to 1.5m. 
These two cases of mass dependant stability and velocity dependant stability are 
clearly energy related. The higher the energy contained in the system, the more 
likely the system is to exhibit unstable behaviour. This boundary has not yet been 
clearly defined. Generally speaking, however, the slower the robot movement across 
the web, the more likely the web is to remain stable given a perfectly symmetrical 
web. Given the levels of CoM displacement, the robot velocity is not a significant 
destabilising factor in this scenario. 

6.8 Statistical investigation into stability 
A series of solutions to the space-web equations of motion were examined to examine 
the effect of five parameters on the space-web stability. The mass of the web, 
Mweb, the mass of the three daughter satellites, Msat, the robot mass, Mrobot, 
the sub-span angular coordinate, . 
and the average sub-span angular velocity @. 


@t 


were thought to have an effect on the maximum movement of the Centre of Mass 
(CoM)of the space-web system. Therefore, the solutions to 36 different sets of ini


162 



tial conditions were found, with 25 =32 runs complemented by 4 centre point runs. 
The 36 runs were performed in Mathematica, and the results of the maximum CoM 
displacement were entered into a statistical package, Design Expert 7.03. Design 
Expert thenperforms an analysis of variance(ANOVA) calculation onthefactorial 
data provided. ANOVA is a collection of statistical models and their associated 
procedures which compare means by splitting the overall observed variance into differentparts. 
A5degreeoffreedommodel(orless, ifrequested) isthenextrapolated 
from the data and analysed for statistical significance. 

The five most strongly correlated model parameters were: 

Mweb- Msat+ Mweb 
* @. 
+ Msat 
* Mrobot 
* @. 
- Mrobot-

@t 
@t 


 + and @. 
- were statistically relevant in the model, but to a lesser degree than 

@t 


the other parameters. As they have a stronger effect when combined with other 
variables,the(un)stabilising influencetheyhavemaybeovertakenby theseother 
factors. 

Parameters areshown with a(+)indicating astabilising effect ora(-)indicating 
a destabilising effect, with the full table of values located in Appendix C. 

As Mweb- has the strongest influence over the model, it would be most beneficial 
to the stability to reduce the mass of the web. 

Thereverse istrueofthedaughtersatellitemass,Msat+, increasingthisparameter 
will tend to stabilise the system. 

For combinations of main parameters such as Mweb 
* @. 
+, it would be most 

@t 


beneficial to increase the product to stabilise the system. In this case, the angular 
momentum of the web will tend to rigidize the system and enhance the stability. 

6.8.1 Masses of the web and satellite 
The influence the masses of the web and the daughter satellites have on the CoM 
positionareshown inthecontourgraph,Figure6.14. Thecontoursareshadedfrom 

163 



blue(stable) throughgreen andyellow to red(unstable) andfollow the maximum 
CoM displacement from the central facility. 

Design-Expert® Software CoM Dist 

10.00 

CoM Dist 


Design Points 
55.7692 


0.00324405 8.00 
X1 =A:Mweb 

10 
15 
20 
25 
30 
35 
40 
45 
4 
X2 =B:Msat 

Actual Factors 

C: Mrobot = 5.50 
D: psi = 0.06 
E: psidot = 0.55 
Msat B: 
6.00 

4.00 

2.00 

27.00 87.75 148.50 209.25 270.00 
A: Mweb 
Figure 6.14: Contours of CoM movement while varying web and satellite mass 

For the small sample space of initial conditions, the most stable point would be a 
high satellite mass(> 
8kg)and low web mass(< 
50kg). A CoM displacement of 
above 10m 
is unstable, and would be very difficult for a robot to manoeuvre over 
the surface. 

Ingeneralterms, this clearly shows the stabilisingeffect ofthe mass ofthe satellites 
and the destabilising effect of the mass of the web. 

6.8.2 Masses of the web and robot 
The influence the masses of the web and the robot have on the CoM position are 
shown inthecontourgraph,Figure6.15. Thecontoursareshadedfromblue(stable) 

164 



throughgreen andyellowto red(unstable) andfollowthe maximumCoMdisplace


ment from the central facility. 

Design-Expert® Software CoM Dist 

10.00 

CoM Dist 

55.7692 

0.00324405 

7.75 

X1 =A:Mweb 
X2 = C: Mrobot 


27.00 87.75 148.50 209.25 270.00 
10 
15 20 
25 30 
35 
40 45 
50 
Actual Factors 

B: Msat = 6.00 
D: psi = 0.05 
Mrobot C: 
5.50 

E: psidot = 0.10 
3.25 

1.00 

A: Mweb 
Figure 6.15: Contours of CoM movement while varying web and robot mass 

The relative strengths of the robot and web masses are shown, with the mass of 
the web exerting a much stronger influence over the behaviour of the web. The 
stability boundary isshown inthebottomlefthand cornerof thegraph. Astability 
margin of 5m 
is preferable, here the web mass must be kept below 50 kg 
and the 
massof therobot,although less influential, wouldbebest suited atbelow5kg. This 
shows that if a low @. 
is required, the web can be made more stable with careful 

@t 


consideration of variables. 

6.8.3 Mass of the web and angular velocity 
The influence the masses of the web and the angular velocity, @. 
, have on the CoM 

@t 


positionareshown inthecontourgraph,Figure6.16. Thecontoursareshadedfrom 
165 


blue(stable) throughgreen andyellow to red(unstable) andfollow the maximum 
CoM displacement from the central facility. 

Design-Expert® Software CoM Dist 

1.00 

CoM Dist 

55.7692 

0.00324405 

0.78 

X1 =A:Mweb 

5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
X2 = E: psidot 
Actual Factors 


B: Msat = 10.00 
C: Mrobot = 1.00 
D: psi = 0.06 
psidot E: 
0.55 

0.33 

0.10 

27.00 87.75 148.50 209.25 270.00 
A: Mweb 
Figure 6.16: Contours of CoM movement while varying web mass and @. 


@t 


In a high-stability scenario, the options to configure the space-web are plentiful. 
Carefully choosing the previous parameters that lead to a high probability of stability, 
namely ahigh satellite mass(10kg)and a low robot mass(1kg), affords a 
larger range of possible values for the mass of the web and the angular velocity of 
the web. 

If the largest acceptable CoM movement is limited to 5m, then there are three 
choices available,depending on theprimary requirement of the space-web system. If 
a low web mass is required, then a low @. 
must match, and vice versa. Alternatively, 

@t 


@. 


if a high mass is required, a high @t 
must be specified, and vice versa. The final 
option is for a low web mass and a high @. 
, affording a large safety margin that 

@t 


may be advantageous in, for example, a pilot study mission. 

166 


6.8.4 Mass of the robot and angular velocity 
The influence the mass of the robot and the angular velocity, @. 
, have on the CoM 

@t 


positionareshown inthecontourgraph,Figure6.17. Thecontoursareshadedfrom 
blue(stable) throughgreen andyellow to red(unstable) andfollow the maximum 
CoM displacement from the central facility. 

Design-Expert® Software CoM Dist 

1.00 

CoM Dist 

55.7692 

0.00324405 

0.78 

X1 = C: Mrobot 
X2 = E: psidot 

1.00 3.25 5.50 7.75 10.00 
5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
Actual Factors 

A: Mweb = 270.00 
B: Msat = 10.00 
D: psi = 0.05 
psidot E: 
0.55 

0.33 

0.10 

C: Mrobot 
Figure 6.17: Contours of CoM movement while varying robot mass and @. 


@t 


The final comparison is for a high mass specification – a 270kg 
web mass and 
10kg 
satellite mass. This combines the two strongest influences on the system, the 
former destabilising and the later stabilising. For a stable system, it is essential to 
ensure the robot mass is small and the angular rotation rate is large. 

167 



6.9 Conclusions and recommendations 
6.9.1 Conclusions 
The achievement of a stable, re-configurable web orbitingin space with robots moving 
along its surface is a realistic goal. There must, however, be limits to the 
behaviour of the robots and the configuration of the web. Both the robots and the 
web must be as light as possible and the robots must be as slow moving as the 
mission allows, given that this limits the destabilising effects of these parameters. 
The daughter satellites must be as heavy as possible, and the angular rotation rate 
must be as large as possible to maximise these stabilising effects. In all cases, the 
web configuration must be as symmetrical as practicable. 

6.9.2 Recommendations 
To capitalise on the knowledge gained in this research, there are a few areas that 
may be expanded on for potential future exploration. 

Constructing the web while on orbit would mitigate the deployment phase of the 
mission, eliminating one of the major failure modes of the system. Using the robots 
to weave the web could be inspired by spiders on Earth, for example, just as the 
space web has been inspired by the Furoshiki cloth. 

Using the knowledge gained through moving masses over the web, there are many 
applications that would benefit from simulating web-based constructions with moving 
masses. Solar-sails and antennae could be assembled or reconfigured by robots 
moving across their surface. 

168 



Chapter 7 

Conclusions 

7.1 Tether rotations 
. 
Rotations using a different system of equations are equally able to describe 
the same system. 

. 
Therefore,thechoiceof rotational systemcanbetailored insuch away tosuit 
the analysis. 

7.2 Tethers with inclination 
. 
A demonstration mission launching a micro-satellite from MEO to Lunar orbit 
is achievable using current technology. The safety margins, however, are 
extremely small and the MMET launcher would be located in an orbit with 
large amounts of debris. 

. 
The inclination term does not significantly alter the dynamics of the MMET 

system. However, any out-of-plane component in the local axes – whether on 

169 



an inclined orbit or not – interacts with the orbital parameters to create an 
unsteady oscillation in the tether. This is unacceptable for payload aiming 
and release, therefore the local out-of-plane angle should be minimised insofar 
as possible. 

7.3 Deployment and recovery of tethers 
. 
Deploying and recovering the tethers on the MMET system is not a trivial 
matter; however, they can be achieved in a well controlled and structured 
way. 

. 
Amethodfordeployment and recoveryintheorbitalplanehavebeenoutlined, 
givingconfidencethattheMMET iscapableand suitableforuseasareusable 
launch platform. 

7.4 Space-webs 
. 
The achievement of a stable, re-configurable web orbitingin space with robots 
moving along its surface is a realistic goal. 

. 
There are limits to the behaviour and configuration of the robots and web: 
the robots must be light and as slow moving, the web must be as light, the 
daughter satellites must be heavy and the angular rotation rate must be as 
large as possible to maximise these stabilising effects. In all cases, the web 
configuration must be as symmetrical as practicable. 

170 



7.5 Further study 
It is my belief that tethers, and in particular the MMET, have a significant role to 
play in mankind’s future in space. By addressing the following issues uncovered in 
this thesis, the case for using the MMET will be strengthened. 

The stator arm will require to be modelled and analysed if the concept of the 
MMET is ever to be successful. The possibility of interference between the tether 
and stator arms will logically follow on from this analysis. 

Likewise, the concept of a space-web is both novel and interesting, and doubtless 
will be studied further. It is essential to perform further ground tests or even zero-
gravity studies of theMMET and space-webstoprovide validation of the equations 
of motion. 

171 



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182 



Glossary 


Notation Description 
LEO Low Earth Orbit 
MEO Medium Earth Orbit 
MMET Motorized Momentum Exchange Tether 
MXER Momentum eXchange/Electrodynamic Reboost 
WSB Weak Stability Boundary 

183 



184 


Appendix A 
Equations of motion with inclination 

Theequationsof motiontheMMET(statorarmnot included) forthegeneralised co-ordinates R, 


 
are: 
R 
equation: 

µMtotal 

+¨ 2 
R2 RMtotal -
R 
cos 2().2 +.Mtotal =0 (A.1) 

. 
equation: 

.

R 
cos() 2cos()R.+ 
R 
cos()
. 
¨ -2.
.sin() MFacility =0 (A.2) 


185 


. 
equation: 

.
2 2.

R2 cos()sin()MFacility+ 
R 
R.+ R¨MFacility =0 (A.3) 

. 
equation: 

. 

. 



1 
..


2cos2()sin(2 ).2 + 4.
a 
sin( )-2 cos( ) .
sin(2)+.2cos(2 )sin()cos2()+cos()cos( )sin(2) .- 

8 

...

4cos( ).
a 
sin()+2. 
sin(2)sin( )sin()+2cos2()sin(2 )sin2()+sin(2)sin(2)sin( ) 

1 

¨ 

(4Mpayload + Mtether)L2 + 
192 
128 
. 
(3Mpayload + Mtether)+ 

48 2cos()
. 
¨cos 2()+2 
. 
¨cos 2()-cos( )
. 
¨sin(2)sin()-2. .
sin(2)+.
(cos()sin(2)+cos(2)cos( )sin()) + 

¨
. 
sin(2)sin( )+. .
sin(2)sin()sin( )+.
. 
cos( ) .
sin(2)-.
2sin()cos2()+cos()cos( )sin(2) +2cos(2).
sin( ) 

(4Mpayload + Mtether))L2 + R
µ 
cos()sin( )
1 
- 
1 
Mpayload+ 

3/23/2

(L2 -2R 
cos()cos( )
L 
+ R2)(L2 +2R 
cos()cos( )
+ R2)

11 

4 - Mtether 


3/23/2

(L2 -4R 
cos()cos( )
L 
+4R2)(L2 +4R 
cos()cos( )
L 
+4R2)

= cosa 
Forcepayload L(1-Eclipse) (A.4) 


186 


a 
equation: 

1 
((6cos(2)+cos(2(
. 
- ))-2cos(2 )+cos(2(
. 
+  ))+2)sin(2)+8cos(2)cos( )sin(2)).2+ 

32 

16  .
(cos()sin(2)+cos(2)cos( )sin()).-8.2 sin(2)sin2( )+8 .
2 sin(2)- 

.

8.
. 
2cos(2)
. 
sin( )+.(2cos(2)cos()sin( )-sin(2)sin()sin(2 )) (4Mpayload + Mtether)L2+ 

. 
. 



128¨(3Mpayload + Mtether)+96 ¨+ cos( )¨
. 
+ cos( ).
 .
sin()+.
. 
cos().- .
sin( )+ 
. 
¨sin()sin( ) (4Mpayload + Mtether) L2+ 

192 

11 

R
µ 
cos( )sin() - Mpayload+ 

3/23/2

(L2 -2R 
cos()cos( )
L 
+ R2)(L2 +2R 
cos()cos( )
L 
+ R2)

11 

4 - Mtether L

3/23/2

(L2 -4R 
cos()cos( )
L 
+4R2)(L2 +4R 
cos()cos( )
L 
+4R2)

= -Forcepayload L(1-Eclipse) (A.5) 

The equations use the following substitutions: 

Mtotal = MFacility +2Mpayload +2Mtether (A.6) 


187 


and Eclipse is the binary function: 

Eclipse = 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
1 
0 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
R2 
. 
1-cos2()cos2() 
. 
= R2 
Earth 
. 
1+ 
R 
cos() 
RSun 2 
AND 
cos()cos()
> 
0 
. 
otherwise 
(A.7) 

Note: the inclination generalised coordinate is 
. 
theninth letterof theGreek alphabet(iota). The firstderivative is @
= . and the 

@


second derivative is @2. 
=¨.

@t2 


Appendix B 
Space-webs – additional graphics 

B.1 
Case1 –CoMplotsof stability whileincreasing 
web mass 
0.3 


0.2 


0.1 
0 
-0.1 
-0.2 
-0.3 


ZCoM Position 
Z


CoM Position 


0.4 


0.2 


Y 
0 


-0.2 


-0.4 


Y 
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 
-0.4 -0.2 0 0.2 0.4 


Figure B.1: mW 
eb 
=27kg 
Figure B.2: mW 
eb 
=33.75kg 


188 



Z


ZCoM Position 


CoM Position 


-0.75 -0.5 -0.25 0 0.25 0.5 0.75 
-0.75 
-0.5 
-0.25 
0 
0.25 
0.5 
0.75 
Y 


-0.4 
-0.2 
0 
0.2 
0.4 
Y 
-0.4 -0.2 0 0.2 0.4 


Figure B.3: mW 
eb 
=37.8kg 
Figure B.4: mW 
eb 
=40.5kg 


Z


CoM Position Z


CoM Position 


3 
2 
1 
0 
-1 
-2 
-3 


15 


10 


5 



Y


0 


-5 


-10 



-15-10 -5 0 5


-3-2-10 1 2 3 


Y 
10 15 20 
Figure B.5: mW 
eb 
=44.82kg 
Figure B.6: mW 
eb 
=47.25kg 


ZZ


CoM Position CoM Position 
40 
30 
20 
10 
Y 


-30 
-20 
-10 
0 
10 
20 
Y 
-10 
-20 
-30 


-40 -30 -20 -10 0 


0 


30 
10 20 30
-30-20-10 0 10 20 30 


Figure B.7: mW 
eb 
=54kg 
Figure B.8: mW 
eb 
=67.5kg 


189 



B.2 Case1 –CoMplotsof stability whileincreasing 
robot mass 
-40 
-20 
0 
20 
40 
ZCoM Position 
Z


CoM Position 



20 


0 


-20 


-40 


Y 


Y 


-40 -20 0 20 40 -40-20 0 20 40 


Figure B.9: Mrobot 
=1kg 
Figure B.10: Mrobot 
=10kg 


Z



CoM Position 
Z


CoM Position 


40 


1.5 
1 
20 


0.5 



Y


Y0 


-0.5 


-20 


-1 
-1.5


-40 
-1.5 -1 -0.5 0 0.5 1 1.5



-40-20 0 20 40 


Figure B.11: Mrobot 
=15kg 
Figure B.12: Mrobot 
=16kg 


ZZ


CoM Position CoM Position 


0.8 



0.2 



0.6 


0.6 


0.4 


0.4 


Y 
0.2 
0 
Y0 
-0.2 


-0.2 



-0.4 


-0.4 
-0.6 



-0.6 
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 



Figure B.13: Mrobot 
=20kg 
Figure B.14: Mrobot 
=100kg 


190 



B.3 CoM plots of stability while changing robot 
paths 
Z


ZCoM Position 


CoM Position 


20 


10 


Y


0 


-10 



-20 


-0.4 -0.2 0 0.2 0.4 


-20-10 0 10 20 


-0.4 
-0.2 
0 
0.2 
0.4 
Y 
Figure B.15: Case 1 

Z


CoM Position 


-15 
-10 
-5 
0 
5 
10 
15 
-15-10-5 0 5 10 15 


Figure B.17: Case 3 

Z


CoM Position 


-30 
-20 
-10 
0 
10 
20 
30 
-40-30-20-10 0 10 20 30 


Figure B.16: Case 2 

Z


CoM Position 
10 
5 



Y


0 


Y 


-5 


-10 
-10-50 510 


Figure B.18: Case 4 

Z


CoM Position 


20 


10 



Y


0 


Y 


-10 


-20 


-30 
-30-20 0 20
-10 10 


Figure B.19: Case 5 Figure B.20: Case 6 

191 



B.4 Case 1 – CoMplots of stability while decreasing 
robot velocity 
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 
-0.3 
-0.2 
-0.1
0 
0.1 
0.2 
0.3 
Y 
CoM PoZsition 
Figure B.21: Vrobot = 10.0m/s 
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 
-0.3 
-0.2 
-0.1
0 
0.1 
0.2 
0.3 
Y 
CoM PoZsition 
Figure B.22: Vrobot = 2.0m/s 
-0.4 -0.2 0 0.2 0.4 
-0.4 
-0.2
0 
0.2 
0.4 
Y 
CoM PoZsition 
Figure B.23: Vrobot = 1.0m/s 
-1 -0.5 0 0.5 1 
-1 
-0.5
0 
0.5
1 
Y 
CoM PoZsition 
Figure B.24: Vrobot = 0.2m/s 
-1.5 -1 -0.5 0 0.5 1 1.5 
-1.5 
-1 
-0.5
0 
0.5
1 
1.5 
Y 
CoM PoZsition 
Figure B.25: Vrobot = 0.1m/s 
192

X 


100 


0 
-50 


-100 
100 


50


50 


50 
-100 
Z 


0 
00

-50 
-50-50

-100 
-100-100

-50 


0 


50 


Y 
100 


Figure B.26: Case 1 – 3 robots moving symmetrically round the web perimeter 

193 



X 


100 


0 
-50 


-100 
100 


50


50 


50 
-100 
Z 


0 
00

-50 
-50-50

-100 
-100-100

-50 


0 


50 


Y 
100 


Figure B.27: Case 6 – 1 robot moving asymmetrically along first sub-span 

194 



Appendix C 
Space-webs – additional data 

Data generated during the first run phase – initially examining the dynamics of 
the space-web. 

The units for the variables are as follows: 

Mweb Msat MRobot psi psidot k CoM disp 

kg 
kg 
kg 
degrees 
radians/s 
N/m 
m 


The initial conditions of all other parameters are: 

L 
=100.0m 
; Mfacility =100.0kg 
; eccent 
=0.0; 
tend =100.0s 
; R 
=6.578* 106 m 
; K 
=122.69 
K 
is the spring stiffness, given in Equation 6.15. This is derived from the Young’s 
Modulus, E 
=3* 1010 and the CSA of each strand of 4 * 10-6 . 
The webs are configured with3 web sections(i.e. 2divisions). 
The angulardisplacements aregiven with one angle only:  . This may be converted 
to give the angles of the three sub-spans such that: {0+ , 
120. , 
240. - }. 
The three angular rates are equal, and are quoted once in the following tables.. 

195 



Datageneratedduring thestatistical investigation intostability(seeSection6.8). 

run A:Mweb. B:Msat C:Mrobot D:psi E:psidot Max CoM 

E01 27. 10 2 1 0.1 0.35 
E02 33.75 10 2 1 0.1 0.41 
E03 37.8 10 2 1 0.1 0.85 
E04 40.5 10 2 1 0.1 0.39 
E05 44.82 10 2 1 0.1 3.74 
E06 47.25 10 2 1 0.1 20.44 
E07 54. 10 2 1 0.1 34.82 
E08 67.5 10 2 1 0.1 39.40 
E09 135. 10 2 1 0.1 46.49 
E10 270. 10 2 1 0.1 51.03 

Table C.1: Influence of web mass on maximum CoM 

run symmetrical A:Mweb B:Msat C:Mrobot D:psi E:psidot Max CoM 

Case 1 yes 40.5 10 10 1 0.1 0.46 
Case 2 no 40.5 10 10 1 0.1 22.61 
Case 3 no 40.5 10 10 1 0.1 18.52 
Case 4 yes 40.5 10 10 1 0.1 10.95 
Case 5 no 40.5 10 10 1 0.1 38.24 
Case 6 no 40.5 10 10 1 0.1 30.99 

Table C.2: Influence of symmetrical robot movement on maximum CoM 

run A:Mweb B:Msat C:Mrobot D:psi E:psidot time Robot speed Max CoM 

FG1 40.5 10 2 1 0.1 10 10.0 0.36 
FG2 40.5 10 2 1 0.1 50 2.0 0.36 
FG3 40.5 10 2 1 0.1 100 1.0 0.40 
FG4 40.5 10 2 1 0.1 500 0.2 1.40 
FG5 40.5 10 2 1 0.1 1000 0.1 1.70 

Table C.3: Influence of simulation time (effectively robot speed across web) on 
maximum CoM 

196 



run A:Mweb B:Msat C:Mrobot D:psi E:psidot Max CoM 

Frobot01 135 10 0.1 1 0.1 45.09 
Frobot02 135 10 1 1 0.1 45.89 
Frobot03 135 10 5 1 0.1 46.66 
Frobot04 135 10 10 1 0.1 46.08 
Frobot05 135 10 12 1 0.1 50.53 
Frobot06 135 10 15 1 0.1 48.44 
Frobot07 135 10 16 1 0.1 1.75 
Frobot08 135 10 17 1 0.1 1.08 
Frobot09 135 10 18 1 0.1 0.86 
Frobot10 135 10 19 1 0.1 0.82 
Frobot11 135 10 20 1 0.1 0.77 
Frobot12 135 10 21 1 0.1 0.66 
Frobot13 135 10 50 1 0.1 0.67 
Frobot14 135 10 100 1 0.1 0.75 

Table C.4: Influence of robot mass on maximum CoM 

197 



run A:Mweb B:Msat C:Mrobot D:psi E:psidot F: CoM posn predicted Diff % 

1 27 2 1 0.01 0.1 11.49 12.29 107% 
2 270 2 1 0.01 0.1 49.41 46.71 95% 
3 27 10 1 0.01 0.1 0.00 -2.70 -83118% 
4 270 10 1 0.01 0.1 52.76 53.56 102% 
27 2 10 0.01 0.1 28.09 25.39 90% 
6 270 2 10 0.01 0.1 53.20 54.00 102% 
7 27 10 10 0.01 0.1 0.00 0.80 19141% 
8 270 10 10 0.01 0.1 55.75 53.05 95% 
9 27 2 1 0.1 0.1 11.28 10.16 90% 
270 2 1 0.1 0.1 49.83 47.37 95% 
11 27 10 1 0.1 0.1 0.03 0.12 369% 
12 270 10 1 0.1 0.1 52.79 51.67 98% 
13 27 2 10 0.1 0.1 27.49 26.72 97% 
14 270 2 10 0.1 0.1 52.99 51.65 97% 
27 10 10 0.1 0.1 0.04 -0.05 -111% 
16 270 10 10 0.1 0.1 55.61 54.83 99% 
17 148.5 6 5.5 0.055 0.55 47.15 31.10 66% 
18 148.5 6 5.5 0.055 0.55 47.15 31.10 66% 
19 148.5 6 5.5 0.055 0.55 47.15 31.10 66% 
148.5 6 5.5 0.055 0.55 47.15 31.10 66% 
21 27 2 1 0.01 1 49.41 46.71 95% 
22 270 2 1 0.01 1 49.86 50.66 102% 
23 27 10 1 0.01 1 0.00 0.80 24649% 
24 270 10 1 0.01 1 0.00 -2.70 -64127% 
27 2 10 0.01 1 33.10 33.90 102% 
26 270 2 10 0.01 1 49.41 46.71 95% 
27 27 10 10 0.01 1 49.83 47.13 95% 
28 270 10 10 0.01 1 55.77 56.57 101% 
29 27 2 1 0.1 1 49.41 48.63 98% 
270 2 1 0.1 1 49.82 48.69 98% 
31 27 10 1 0.1 1 0.03 -1.09 -3346% 
32 270 10 1 0.1 1 0.03 -0.74 -2287% 
33 27 2 10 0.1 1 33.11 31.98 97% 
34 270 2 10 0.1 1 49.41 48.63 98% 
27 10 10 0.1 1 0.00 -0.77 -18329% 
36 270 10 10 0.1 1 55.76 54.63 98% 

Table C.5: Statistical investigation into stability 

198 



Term Effect Stdized Effect SumSqr % Contribtn 

A-Mweb -27.21 7182.9 37.72 
B-Msat + -16.57 2447.8 12.85 
AE + -15.32 1869.6 9.82 
BCE -14.34 1656.8 8.70 
C-Mrobot -10.90 872.3 4.58 
BC -9.93 666.4 3.50 
BE + -8.69 686.3 3.60 
AB -7.21 435.0 2.28 
ACE -6.98 346.3 1.82 
CE -5.04 232.9 1.22 
AC -4.66 193.2 1.01 
ABE + -3.76 118.3 0.62 
ABC -3.75 118.6 0.62 
BDE + -3.37 93.4 0.49 
ADE -3.29 117.6 0.62 
ABCDE -3.19 85.1 0.45 
ACD -3.17 99.5 0.52 
BCD + -3.15 102.7 0.54 
CDE + -3.12 92.4 0.49 
ABDE -3.12 81.5 0.43 
CD + -3.09 110.1 0.58 
D-psi + -3.08 79.6 0.42 
BCDE + -3.08 83.7 0.44 
ACDE -3.04 89.0 0.47 
ABCD -3.02 61.5 0.32 
DE + -3.00 75.6 0.40 
ABD -2.97 74.0 0.39 
AD -2.93 98.7 0.52 
BD + -2.84 34.2 0.18 
E-psidot -1.48 18.3 0.10 

Lack of fit 9.6 0.05 
Pure error 0.0 0.00 


Table C.6: Results of the statistical investigation into stability 

The ‘Effect’ term incolumn2 isareflectiononthestabilising(+) ordestabilising 
(-)effect of the term. 

199 



Appendix D 

Material strengths 

The information in Table D.1 is provided courtesy of the Island One Society at 

http://www.islandone.org/LEOBiblio/SPBI1MA.HTM, and is compiled from data 

from the following sources: [Brown, 1989, pp 14-18], [Rosato, 1993, p 638], [Weast, 

1985, p E-43]. 

1aramid fiber 
2gel-spun polyethylene 
3gel-spun polyethylene 
4plastic fiber, Zylon is brand name of PBO 
5theoretical data 


200 


Material Property 

tensile Young’s density () characteristic speed of 
strength 
(P) 
modulus 
(Y) 
velocity 
. 
2P/. 
sound in thin 
rods . 
Y/. 
[GPa] [GPa] [kg/m3] [m/s] [m/s] 

steel 1-5 200 7900 503-1125 5032 
aluminum alloys 0.1-0.7 72 2700 272-720 5270 
titanium alloys 0.6-1.3 110 5000 490-721 4690 
berylium fiber 3.3 310 1870 1879 12870 
boron fiber 3.5 400 2450 1690 12778 
fused silica 73 --2200 5760 
pyrex glass 62 --2320 5170 
E-glass fiber 2.4 72.4 2540 1375 5339 
S-glass fiber 4.5 85.5 2490 1901 5860 
Kevlar 491 3.6 130 1440 2236 9502 
Spectra 1000 fiber2 3.0 170 970 2487 13239 
Spectra 2000 fiber3 3.51 -970 2690 -
PBO (Zylon)4 5.8 280-365 1560-1580 2710-2727 13397-15199 
carbon fiber 1-6.5 250-830 1850 1040-2651 11600-21200 
buckytube cable5 150 630 1300 15191 22014 

Table D.1: Properties of selected materials 

201 



Appendix E 
Motor properties 

Specification 

List Price: $910 
Horsepower 4hp 
RPM 4150 
mass 23kg 
Armature Volts 48V 
Torque 6.86Nm 
Full Load Current 72.0A 


Table E.1: Properties of GE Motor 5BC49JB1115 

The information in Table E.1 is provided courtesy of the GE industrial products 
guide at: 
www.geindustrial.com/cwc/marketing/Motors/catalog/CrossReference.pdf 
and is compiled from data from the following additional source: 

www.emotorstore.com/productdetail.asp_Q_brandID_E_1_A_catID_E_23_A_ 
subCatID_E_354_A_productID_E_583_A_skuID_E_30698 


The motor is a GE Industrial Motor, Product Number D379, Model Number 
5BC49JB1115. ThisisaDC motor, used topower electric vehicles(commonlygolf-carts). 
The specifications are used in Section 4.5.1 to gain an approximate mass, torque and 
rotational speedforan equivalent space-based motor. Thetorqueisquoted as81.0oz-ft, 
and converted to Nm. 

202