A review of space tether research

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Progress in Aerospace Sciences 44 (2008) 1–21 www.elsevier.com/locate/paerosci

A review of space tether research M.P. Cartmell, D.J. McKenzie Department of Mechanical Engineering, University of Glasgow, James Watt Building, Glasgow G12 8QQ, Scotland, UK Available online 7 November 2007

Abstract The review paper attempts to provide a useful contextualised source of references for the student interested in learning about space tethers, and their potential for propulsion of payloads in Space. The two principal categories of momentum exchange and electrodynamic tethers are discussed, with the principal aim of establishing useful sources of fundamental theory in the literature, as well as highlighting important technology and mission development papers. The large-scale international effort that continues to be made in the area of space tether research is evident, with major literature contributions from the world-wide scientific and technical community. The overarching theme of the paper is to show the richness and diversity of tether modelling that has been undertaken in recent times, and to emphasise, by means of many different examples, that dynamics and control are the two fundamentally important aspects of all tether concepts, designs, and mission architectures. r 2007 Elsevier Ltd. All rights reserved.

Contents 1. 2.

3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Momentum exchange tethers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1. Summary of operating principles and relevant orbital mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2. Tether missions, constraints, and failure modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3. Dynamics of dumb-bell systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4. Tether models in which flexural effects are introduced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5. Control strategies and models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.6. Practical tether designs and proposed system technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.7. Deployment scenarios and mission plans. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Electrodynamic tethers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1. Summary of operating principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.1. The TSS-1R mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2. Practical electrodynamic tether designs and proposed system technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1. Introduction The field of space tethers has received very considerable attention in recent decades, with many specialist articles Corresponding author. Tel.: +44 141 330 4337; fax: +44 141 330 4343.

E-mail address: [email protected] (M.P. Cartmell). 0376-0421/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.paerosci.2007.08.002

available in the technical and scientific literature. Some of these are reviewed in this paper, and the discussion also covers some of the texts and handbooks available. We start with the excellent foundation textbook by Beletsky and Levin [1] in which the dynamics of tethers are introduced rigorously, in a progressive and pragmatic manner. The book starts by setting the scene for tethers in space by

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summarising possible applications and also by discussing fact and fiction in such a way that the reader new to tethers is soon clear about the important fundamental parameters such as material density, strength, and orbital location, and how these can trade off. The book then moves on to develop equations of motion for a flexible tether with end masses and massless and massive variations, along with perturbational and certain environmental effects. The tether is then investigated within the Newtonian field and the dynamics examined in terms of stability and oscillatory behaviour. Atmospheric probes, electrodynamic (ED) tethers, libration and rotation, deployment and retrieval, and lunar anchored and satellite ring systems serve to complete the coverage of the book. A very useful set of references is also provided, up to the publication year of 1993. A chapter on the orbital mechanics of propellantless propulsion systems, by McInnes and Cartmell is given in the more general astrodynamics text [1], and this covers both solar sails and tethers, as technologies with the potential to overcome the constraints of propulsion based on reaction, and in the tether application this is by momentum balance through the system. The chapter reviews some of the more well-known missions that have flown to date and then moves on to summarise the performance expectations of hanging, librating, and spinning tethers, setting them in the context of results extant in the literature. Gravity gradient stabilisation is re-examined, and the well-known literature result for sub-span tension for a short hanging tether on a circular orbit is obtained and re-cast into the notation of [1]. The motorised tether concept is introduced next, and the equation of motion for a simple motorised dumb-bell on a circular orbit is derived, leading into the more general non-planar case. Payload transfer concepts, including the use of staged tethers (crossreference with [2–4]) are discussed and further useful references are cited. There have been several general short article expositions of tether technology during the last few decades appearing in widely different areas of the literature, starting with a particularly accessible and notable example from Bekey [5]. In this summary discussion Bekey gives some of the history of the subject, with good reference to missions up to 1983 and those planned for a few years after. Principles of momentum exchange and electrodynamics are outlined and useful data is provided. This paper also discusses speculative applications for cryogenic propellant storage and transfer, two-dimensional tethered constellations, the construction of a passive space facility in which platforms are separated by tethers giving a possible work volume within, payload orbit raising and lowering, and a two-tether elevator for transfer from LEO to GEO. In a similar vein Carroll’s paper of 1985 [6] also sets out the history of space tethers with a useful, applications orientated, introduction to the theory in which the important Lunavator concept of Moravec [7] appears, with this further explained and applied by Forward [8], and applied again by Cartmell and Ziegler [9], noting also the more recent summaries given in [10,11]. It is also interesting

to consider tethers acting as tension members within solar sail structures, and also applied as links between highaltitude sails and lower-altitude payloads. Further work on complex tethered systems has led to the notion of the space web, where multiple tethers are set up to comprise an intricate web-like structure, McKenzie and Cartmell [12] and McKenzie [13]. Carroll’s paper [6] also highlights aerodynamic applications where a tethered balloon could exploit atmospheric braking to lower a higher-altitude space plane, and altitude ‘juggling’ whereby a sortie vehicle is raised and lowered by means of a local closed orbit controlled by a variable length-spinning tether. Carroll [6] also introduces intriguing concepts of momentum transfer with celestial bodies, which are examined a little further later on. The major contribution to space tether research made by Robert L. Forward cannot be overestimated, and a short and digestible article by Forward and Hoyt in Scientific American in 1999 [14] 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; note also the later contributions of [4,10,15]. US mission plans for tethers were reviewed in 1999 and summarised in a short paper by Johnson et al. [16], in which the use of ED tethers for reusable upper stages was to be demonstrated during the ProSEDS mission. This had been planned as a conductive tether, for electromagnetic orbital adjustment, approximately 15 km in length with 10 km of it insulated and the remaining 5 km as bare conductor. The mission was scheduled to be flown along with a launch of a Global Positioning System satellite in spring 2003, but was ultimately cancelled, initially because of concerns about potential collision with the ISS. An interesting comparative study of in-space propulsion performance for 1 year crewed Mars mission in 2018 was reported in 2001 by Rauwolf et al. [17]. This comprehensive study undertook to examine contender propulsion technologies: chemical, bimodal nuclear thermal rocket, high-power nuclear electric, momentum tether/chemical hybrid, solar, solar/chemical, and variable specific impulse magnetoplasma rocket (VASIMR). This paper concluded that tether-based mission scenarios looked ‘attractive’ in terms of performance but the two concerns of technology immaturity and operational complexity militated against a projected 2018 application. This important conclusion certainly underlines the need for continued international effort in all aspects of tether science and technology. Also in 2001 a major new proposal emerged as a result of a US-based research collaboration led by NASA Marshall Space Flight Center, and was reported by Sorensen [18]. This paper introduced an ingenious concept in which both momentum exchange and ED reboost could be used for propellantless orbital transfer. This Momentum eXchange Electrodynamic Reboost (MXER) principle relies on the tether rotating as it travels on an elliptical orbit, catching a payload in LEO, and then transferring this to a higher orbit after, say, one period. The electrodynamics would be used to re-boost the

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tether over a period of several weeks prior to the next cycle. This has the potential for a high-performance orbital transfer with essentially free re-boost, and the Final Report [19] of 2003 shows that survivability and flight validation issues are of primary importance, but that the necessary science base and the basic contributory technologies are more or less in place for mission development to continue. In 2006, Bonometti et al. [20] confirmed that MXER continues development within the NASA In-Space Propulsion Technology (ISPT) programme; note also Ref. [21,22]. Tethers and debris mitigation were brought to the fore in 2001 in a useful paper by van der Heide and Kruijff [23], in which limitations of use are defined particularly for deorbiting applications in terms of susceptibility to orbital debris. This paper also discusses the potential problem of tether–tether collisions with a suggestion that when considering de-orbiting missions then 40 constellation deorbits per year corresponds to approximately 4 tethers in space at the same time, which could feasibly be coordinated in order to avoid tether–tether collisions. Further work on tether survivability has been carried out by Draper [24] in which a range of statistical methodologies for life prediction from the literature were critically compared and a revised proposal made, with some potential for practical use highlighted. In addition to the environmental effects of debris and other degrading phenomena tethers and associated payloads are extremely vulnerable to destabilisation because of astrodynamic and other perturbational effects. Practical application of any tether system in space requires a high level of stability control, and the stability of a spinning generalised satellite acted upon by the gravity gradient and constant torques is examined by Sarychev et al. [25]. Importantly, it is shown in this paper that stable equilibria can exist for many general values of the inertial parameters of the satellite. There is a large literature on tethers, and the subset of this which deals with dynamics and control is also substantial, with many models proposed, together with a very large number of associated analyses of potential dynamic performance. Dumb-bell models tend to proliferate, for a range of different momentum exchange configurations in which the tether and payload system is assumed to behave predominantly as a rigid body. Whilst this is a somewhat questionable assumption, and certainly not the case for all phases of deployment and operation, it maybe has some merit for initial studies of new ideas, and many numerical results are obtainable in the literature for such models, some of which are cited here. A paper by Cartmell et al. [26], dealing with applications of the multiple scales perturbation method to weakly nonlinear dynamical systems, proposes approximate analytical solutions for relatively simple dumb-bell models. Numerical solutions to these, and allied models, are discussed in Section 2. The remainder of the paper is divided into two main sections, dealing with momentum exchange tethers and ED tethers, and then completing with some conclusions and a list of 122 references. The momentum exchange discussions in Section 2 offer a

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summary of operating principles and relevant orbital mechanics, then a sub-section on tether missions, constraints, and failure modes. After that the dynamics of dumb-bell systems, tether models in which flexural effects are introduced, control strategies and models, practical tether designs and proposed system technologies, and finally deployment scenarios and mission plans are all investigated and commentaries provided on each topic. The third section on ED tethers is split into two sub-sections, offering a summary of operating principles and one of the principal missions carried out to date, and then practical tether designs and proposed system technologies. 2. Momentum exchange tethers 2.1. Summary of operating principles and relevant orbital mechanics Eiden and Cartmell [27] have summarised briefly the possible role of a European roadmap for non-conductive tethers, nominally based on momentum exchange, and also for conductive tethers in which electrodynamics, and possibly momentum exchange, provide propulsion. In the case of the former class small and large payload de-orbit are seen as near term goals, with free-flying tethered platforms and 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 case for long momentum exchange tethers. The work by Cartmell et al. [26] considers dumb-bell models 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 (refer to Section 2.7 for more on this theme, particularly [11]). An analytical solution for planar librations of a gravity stabilised satellite by Hablani and Shrivastava [28] shows that a perturbational type of twoterm solution can be developed to predict the pitching librations of an arbitrary gravity stabilised artificial rigid satellite in an eccentric orbit. This followed over a decade of extensive international work on this type of problem and the results in [28] show that periodic responses for librating systems are necessarily important and form the spines of the systems’ integral manifolds [29]. Gravity gradient stabilisation of tethers is discussed in depth in [1,10] and features explicitly and implicitly in a very large number of publications in the field, many of which appear here in this review. An important paper by Kyroudis and Conway [30] considered the propulsion advantages of using an elliptically orbiting, tethered dumb-bell system for geosynchronous satellite transfer over the conventional non-tethered impulsive Hohmann transfer. This was done by forming the planar equations for the system and solving them numerically, notwithstanding that the analysis neglected the tether mass and assumed dissimilar end masses in the

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form of the space shuttle at one end and a satellite payload at the other. In general tether propulsion performance was found to improve by using a long tether and a highly eccentric orbit, and this mode showed significant improvements over a reasonably comparable Hohmann transfer. Further work on using tether-based transfers was reported by Lorenzini et al. [2] in their landmark paper in which staged tethers in resonant orbits are proposed as being more mass efficient than single tether systems, with a mass ratio of 1:3 using current materials. Earlier work on staged tethers is usefully summarised in [3] by Hoyt and Forward. Lorenzini et al. [2] briefly refer to tether orbit raising results cited by Carroll [31] for radial separation as a function of tether length, and conclude that spinning staged tethers could provide an ideal transfer rate of five transfers per year. The transfer rate of a staged 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. Orbit raising predictions for tethers are discussed further in Section 2.7 [11]. Continuing with the theme of propulsion of a small payload tethered to a large mass in the form of a space station or large shuttle, Pascal et al. [32] investigated the laws of deployment and retrieval by means of a threedimensional rigid body model of a dumb-bell tether in both circular and elliptical orbits. 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, for which control laws can be stipulated. It is shown that deployment is generally stable whereas retrieval is not. Various laws are examined for deployments 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 Danilin et al. [33] in 1999, in which the elastic tether model of No and Cochrane Jr [34] is used but with different variables and derivation. The objective of this paper 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 as a series of discrete masses interconnected by massless elements and with internal viscous damping. The authors make the very 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 3 km tether, with elemental spacing of 100 m, on orbit. 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. It is also possible, and potentially very useful, to consider tethered vehicles within an aerobraking context. In such cases a vehicle, or probe, and an orbiter, connected together by a tether, are configured so that the vehicle passes through a planetary atmosphere to obtain a target velocity change, with the orbiter passing above the atmospheric influence. Longuski et al. [35] give a full account of this very interesting problem. Their modelling is based on the premise that a dumb-bell tether arrives spinning retrograde to the orbit and when the lower payload enters the atmosphere the aerodynamic effects decelerate the tether until it reaches a minimum orientation angle at which point the drag starts to spin the tether in the opposite direction. An optimisation scheme referred to as spin matching is used to equate the spin rates entering and leaving the atmosphere. This has the desirable effect of minimising the forces on the tether during the manoeuvre. They consider the atmospheric fly-through as an impact problem and the analysis is configured to lead to conclusions for mass optimisation, with gas giants such as Jupiter used for the environmental context. It is shown to work well for massive tethers interacting with the Jovian atmosphere, and the results are particularly tractable in that they only require knowledge of four parameters for massive tethers (orbiter-to-probe mass ratio, non-dimensionalised clearance between minimum altitude of the orbiter and minimum altitude of the probe, non-dimensionalised speed, and DV). In the case of smaller tethers the DV is subsumed within a revised non-dimensionalised speed variable, so the parameter space is reduced to three. A major and authoritative work on the dynamic analysis of tethers using continuum modelling has been provided by Auzinger et al. [36], in which stiff equations of motion obtained by Hamilton-Ostrogradskii and balance principles are solved numerically in a detailed parametric study. This sophisticated numerical investigation offers a great deal of useful predictive data on momentum exchange systems. It should be pointed out that the papers cited above also contain valuable sources of references, some of which also feature in this review, and the interested reader is strongly advised to consult widely on each sub-topic, using the references supplied within this review and also those which are precluded from inclusion due to space reasons but which can be found from the cited papers. In addition to introductory issues of performance, orbital contextualisation, modelling strategy, deployment, and aerodynamic effects, it is also important to appreciate that collision prevention necessarily features within any serious applications for tethers and we introduce some of the literature on this and related matters next. An interesting introduction to calculating collision probability between a tether and a satellite is given by Patera [37,38] of the Center for Orbital and Reentry Debris Studies in Los Angeles, based on a computational scheme for long slender tethers of predefined shape and a spherical collision space on the basis of

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respective state vectors and error covariance matrices. The problem is shown to reduce to a two-dimensional symmetric probability density over a cross-sectional collision region. This in turn reduces to a one-dimensional path integral that gives computational efficiency. A reasonable tether length of 20 km was assumed, with negligible radius and the highly significant conclusion was that tether–satellite collision probabilities are up to 600 times greater than those for satellite–satellite collisions. Useful comparison data is given to support this conclusion in tabular and graphical form for different, and practically feasible, tether configurations. In addition to tether–satellite collisions we also have to consider the susceptibility of tethers to debris impacts. A novel Tether Risk Assessment Programme (TRAP model) due to Gittins et al. [39] has been proposed and comprises three main functions; the breakup function, the tether function, and the analysis function. This work was motivated by the widely held belief that on impact a debris fragment with a diameter a little smaller than half of a tether strand diameter can cause that strand to fail. The break-up component models collisions and explosions to determine fragment number, mass, diameter and DV [40]. The tether function is in fact a model of tether dynamics and this is a two stage affair, dealing with system centre of mass motion and also libration of the end masses and tether mass beads. The analysis function determines collision and severance risk based on probabilistic continuum dynamics in which an orbital trajectory is found between two position vectors when the time of flight is known; this is the well-known Gauss-Lambert problem. The paper considers a single strand tether and, interestingly, a double stranded tether design, where the failure criterion is if both strands in one segment fail or if one mass bead is hit directly. General conclusions were that a two-strand tether has a severance risk of two orders of magnitude lower than the single strand case, with obvious implications for multistrand designs, noting that this premise is also discussed in some detail in [24]. The design of tethers for survivability is revisited in Section 2.5 (see [41]) at which point the patented HoytetherTM is summarised. This multi-line concept has a multi-decade lifetime prediction. From the perspective gained up to this point it is now relevant to introduce orbital injection and basic mission requirements. For the purposes of introduction we consider hyperbolic injections [21], periodic solutions and the control of tethers in elliptical orbits [42], and the all important problem of catching a spacecraft or payload with a spinning tether [43]. Sorensen [21] provides a highly readable account of the issues surrounding the orbital dynamics of the ingenious MXER tether design (also see [19,22]). A long, 100 km or so, high-strength conductive tether uses momentum exchange to catch a payload and then release it into a higher-energy orbit, and then electrodynamics are employed to reboost the tether; effectively to restore energy and momentum given to the payload. Note that ED tethers are considered in more detail in Section 3.1. Sorensen confirms that interplanetary flights require orbits to be configured for hyperbolic Earth

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escape trajectories. It is pointed out that there is flexibility in this because there are a number of hyperbolae whose outgoing asymptotes are identical. The objective is to secure a hyperbolic injection that has an equatorial periapsis, and a methodology is given in [21] to obtain the necessary orbital elements. Appreciation of the whole dynamic context is important for tether mission development and Takeichi et al. [42] provide a control scenario for a rigid body dumb-bell tether in an elliptical orbit. The equations of motion are solved for libration and it is shown that the total energy of the system is minimised when the librational and orbital motions coincide with periodic solutions. The overall conclusion is that the periodic solution is of minimum energy and that this minimum is the case for circular or eccentric orbits as long as the libration is actually possible. This can be assured through a simple periodic on-off control strategy at a certain true anomaly. The usefulness, or otherwise, of tether libration is revisited by Ziegler and Cartmell [11] but clearly it has the potential for distinct advantage over the hanging configuration for payload increment gain. It is equally obvious that spinning tethers have greater potential still [11]. On the assumption that we can design long-lasting tethers, configure them into suitable orbits, control their dynamics for optimal payload propulsion and minimal potential for collision, the next step is to introduce criteria for efficient rendezvous with spacecraft and payloads. Lorenzini [43] provides an in-depth treatment of a spinning tether loop with an extended time opportunity for error-tolerant payload capture within high DV propulsion to GTO and Earth escape. The configuration is such that the ends of the loop are furthest away from the centre of mass, where the loop is at its narrowest. The concept is simple in principle, depending on the ejection of a payload (satellite) located harpoon that hooks onto the loop. 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. Stabilisation of post capture oscillations would be required prior to further release into a higher-energy orbit. The author rightly points out that other analyses would be required to address loop deployment, fault tolerance, and mitigation of entanglement. A useful survey of the dynamics and control of tethers is given by Misra and Modi [44]. Cartmell and D’Arrigo [45] modelled a symmetrical motorised momentum exchange tether with manipulation of the payload endmasses in order to generate inertial parametric excitation of the system, in order to determine if the forced-parametric bifurcatory states could be used to guarantee monotonic spin-up, with preliminary results indicating that this is indeed possible. The interested reader will find useful crossreferences available from [4,5,46]. 2.2. Tether missions, constraints, and failure modes One of the fundamentally important issues surrounding successful tether flight is that of the avoidance of

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entanglement and collision involving these long and vulnerable structures. This was investigated by Chobotov and Mains [47] for the TSS-1R mission, which was flown in conjunction with the space shuttle orbiter STS-75 in February 1996. The paper gives a summary of the history of the TSS missions in 1992 and 1996 and concludes that the TSS-1R mission deployed to 19.7 km instead of the planned 20 km with a failure of this ED test tether due to a foreign object penetrating the insulation layer which then exacerbated failure due to arcing and burning of the tether at a nominal tensile load of 65 N. The paper by Chobotov and Mains [47] offers an interesting study into the probability of TSS collisions with micrometeoroids and other orbiting objects. The authors state that the expectation was that the extended tether would be ‘expected to be impacted by multiple particles 0.1 mm or larger in size’ and that ‘the probability of collision with objects 10 cm or larger in size was small (10 3) before the TSS re-entered and burned up in the atmosphere 3 weeks after deployment from the shuttle’. The NASA EVOLVE model for manmade debris flux was used as the basis for this work, at a 350 km altitude. In addition to this data for the closest distance to satellites in this location was also used, this being based around the US Space Command SGP4 propagator, and a simulation for a six month period starting 1 March 1996 generated over 58 000 objects and 24 satellites. The results from the analytical model were compared to those from a statistical Weibull probability density function approach, with good mutual correlation. In the case of the TSS study it was concluded that the tether was subjected to several impacts by small particles greater than 0.1 mm in size and that the probability of collision with larger orbiting objects was very small, and of the order of 0.001 per month. This paper provides a useful set of methodologies for general application to tethers. The subject of tether failure, particularly at deployment, was further reviewed, by Gates et al. [48], and in the context of the Advanced Tether Experiment (ATEx), launched in October 1998, which is particularly interesting because the tether used had a flat tape-like cross-section. The deployment failed after only 22 m out of the possible 6.5 km length, and the mission was aborted. The tether was made from low-density polyethylene with three SpectraTM reinforcing strands running lengthwise along the tether and showed a tendency to stick to itself and for this selfadhesion to increase with stowage time. It also exhibited a mechanical memory effect. The mission was aborted because the deployment problems led to an excessive libration of the deploying tether outside the acceptable location boundaries. The paper by Gates et al. [48] succinctly describes the build-up to this automatic system decision and also goes into useful practical detail regarding the deployer and the deployment methodology. The intended deployment rate was quite low at 0.02 m/s over 3.5 days, with built in accommodation of gravity gradient-induced librations. The expectation was that as the deployment progressed the libration angle would

reduce, partially by means of changing gravity gradient effects during the deployment and partly because of openloop spacecraft manoeuvres. A key sensor, measuring tether position in a plane at the top of the lower tethered body, and relative to the axis through this, showed slackness, and this led to an automated signal to jettison. The actual cause of the excessive slackness was not detectable due to limited telemetry. Interestingly after ruling out certain failure modes the authors propose that the most likely failure mode was thermal expansion. This seems to have been due to beginning the deployment in eclipse, with telemetry nominal, until the tether entered the sunlight. Another possible cause was the build up of static which may have interfered with the telemetry and generated a false slack signal. Shape memory and tip-off dynamics have also been mentioned as contributory factors, but the thermal expansion effect was deemed to be the most likely cause, possibly in concert with these and other subsidiary effects on top. The paper concludes that tether deployment requirements are extremely important when designing a system for flight, and that large design margins are needed, but were certainly not available in this rather tightly constrained example. Another case study paper is that due to Leamy et al. [49] in which the authors consider the ProSEDS mission and two different finite element simulation models for the dynamics and a fuzzy set technique for the ED and deployment operation, particularly with regard to parameter sensitivity. The paper starts with a short, but useful, summary of tether flights to date (2001) and mentions a few of the futuristic applications that have been proposed. Unfortunately the ProSEDS mission was cancelled in October 2003. This was partly because of a drastically reduced starting altitude and a launch timeframe during a period of solar minimum, which led to the available ED propulsion performance of the ProSEDS system becoming insufficient to meet the mission objectives. Despite this the paper is interesting and highly relevant as a case study. The idea of this mission had been to study bare-wire ED tethers, and to include a thermal model accounting for radiated heat, solar heating, and ohmic heating in order to calculate tether element temperatures, and an atmospheric and planetary model accommodating aerodynamic drag, electron density at each element position, and magnetic field properties. The authors used the two finite element codes to simulate the deployment and electrodynamics of the ProSEDS mission and variable numbers of nodes were reckoned to be superior in this context to fixed node number calculations. The overall findings were that the ProSEDS ED tether operation would not have been particularly sensitive to variation in the material parameters, but initial ejection tether momentum and controller parameters would have been significant, as would variations in the geomagnetic field and the plasma parameters. This clearly indicates once again the importance of controllable and adaptive deployment and braking strategies. It has been shown in this brief review of papers dealing with possible failure and

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constraint modes that successful use of tethers requires a good understanding of the environmental conditions both in terms of the prevailing orbital mechanics and the debris environment, a precise knowledge of neighbouring satellite and spacecraft operations, extremely good mechanical design particularly for deployers and brakes, bearing in mind that tether geometry characteristics are an integral part of this, robust sensing of all significant operational variables, and highly adaptive control strategies. These themes are revisited further in subsequent sections of the paper. 2.3. Dynamics of dumb-bell systems A great many tether systems can be considered as some form of dumb-bell system in which two massive bodies, not necessarily of the same mass or size, are coupled together by a low-mass tether by which momentum is exchanged between them. Kelly [50] provides a general overview of possible applications for the conventionally expendable Space Shuttle External Tank (ET) as a space platform in LEO for use as an extended on-orbit crew or experimental package base, or as a micro-g experiment and processing facility, a celestial observation facility, an Earth observation point, or as a staging point on the way to geosynchronous orbit locations. Conceptualisations of these scenarios are provided but it is the apparently secondary roles for the ET that could well involve a dumb-bell tether configuration. Kelly suggests three scenarios, one of which uses a tethered release of the ET from the orbiter in which momentum exchange leads to a boost to a higher orbit for the orbiter and consequently a deorbiting of the ET. This idea, and several others, is also given by Beletsky and Levin [1]. In 1992 Kumar, Kumar, and Misra [51] presented their findings on the effect of deployment rate and librations on tethered payload raising. This seminal paper showed that a counter intuitive result was obtained whereby increasing deployment rate does not necessarily lead to increased payload apogee. They introduce a special rule for planar librations and circular pre-release orbits which they denote as the ‘7+4d’ rule. Additionally the paper shows clear general relationships between apogee altitude gain as a function of deployment rate and explains how suitable deployment rates could be selected for optimising altitude gain, for a given system. The ‘7+4d‘ rule is revisited in [11]. A relatively short class of dumb-bell tether belongs to the OEDIPUS ionospheric plasma test mission system comprising two axially spinning sub-payloads separated by a tether of up to 1 km in length. This was reported by Vigneron et al. [52] in 1997. Terrestrial tests were performed with the understanding that they would differ from the in-space configuration in terms of length scale, higher terrestrial level of gravity, and the presence of friction within the system-supporting gimbals that would not be present in space. The authors derived a specific mathematical model of the laboratory system, one that included the terrestrial effects as well as

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the in-flight phenomena. The paper shows that the model for this problem can be reduced to a linear, vibratory, damped, and gyroscopic system, for which an eigenfunction analysis is used to obtain the damped gyroscopic modes shapes, stability, and natural frequencies for various physical configurations. Interestingly, this work showed that linear modelling could be used to represent modal frequencies and payload attitude stability quite well, however, it obviously did not cover all the possible dynamical phenomena in the system, and would overlook certain regions of convergent attitude motion and limit cycle behaviour. By moving the spin axis so that it is normal to the tether and half-way along the tether length, orientating the tether spin plane so that it is coplanar to the orbit plane, and then forcing the system by means of an external drive motor, a motorised dumb-bell tether can be envisioned. This was first presented in 1998 by Cartmell [53] and a preliminary model was established which showed that forced, motor driven, spin could be generated for a large symmetrical dumb-bell tether, and that complicated non-planar motions of the tether could also be initiated. Motorised tethers are examined further in later sections [4,11,13,54–56], where it is shown that there can be certain advantages to employing an additional form of energy input in this way, notwithstanding the potential for complicated three-dimensional motions as this couples with the prevailing orbital mechanics. Important aspects of the dynamics, which underpin the general stability and control problem that exists with long tethers in space were examined in 2000 by Kumar and Kumar [57] in a system comprising four equal, but short, tethers joining two spacecraft platforms, or satellites. A stability criterion is evolved for a somewhat simplified situation using first order perturbation equations around the nominal equilibrium configuration. The set of rather complicated ordinary nonlinear differential equations is non-dimensionalised and the reduced parameter space is numerically explored. Ultimately the authors consider three configurations; four parallel equi-spaced tethers, a ‘parachute’ arrangement where the four tethers are spaced out at the upper end but converge to a common point on the lower satellite, and a single tether. This intriguing paper shows that all three configurations can provide augmentation of gravity gradient stabilisation, with the parachute layout performing best of all. The tether lengths are extremely low, just a few metres, and it should be emphasised that the objective of this particular paper is to show how very short tethers can be used to give a high degree of threedimensional librational stability to medium-sized systems. Clearly this is a very different remit to the needs of interplanetary propulsion using momentum exchange tethers, but serves to show a useful and very interesting additional application. A discussion of an artificial gravity system, which comprises two tethered satellites was given by Mazzoleni and Hoffman [58] in 2003 and includes tether elasticity within the so-called Tethered Artificial Gravity (TAG) satellite. Tethers are useful for artificial gravity

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generation because they can be used to maximise the r and therefore minimise the o within the o2r that defines the socalled ‘g-force’. The spin-up phase is examined in particular, and it is found that an initial out-of-plane angle of the system and the location of the tether attachment point can both significantly affect the dynamics of the endbody motion of a tethered satellite system (TSS) during spin-up. The modelling included tether elasticity and was based on relevant work reported by Kumar and Kumar [57]. The net conclusion is that if tethers are to be used successfully for artificial gravity generation then attitude control of the end bodies is required during spin-up. Mazzoleni and Hoffman investigate the non-planar spinup dynamics of the Advanced Safety Tether Operation and Reliability (ASTOR) satellite in [59] and show that this spin-up manoeuvre is an example of artificial gravity, which could perhaps be harnessed within human-based missions in the future. Applications of tethers for interplanetary payload propulsion are quite numerous within the literature and some cases have been summarised above in various contexts. Nordley [60], preceded by Forward and Nordley in 1999 [61] (see Section 2.7), showed in 2001 that a dumbbell tether, with a counter mass at one end and a payload at the other, could be used to throw substantial payloads to Mars. This paper concentrates on a digestible summary of the mission architecture strategy necessary to accomplish this and necessarily omits some of the details. However, it is of interest as a pragmatic assessment of the capabilities of a momentum exchange tether on the basis of some reasonable simplifying assumptions using currently available materials and technologies. The general finding from this work is that spinning tethers could be used to propel sizeable payloads to Mars for the same or less total mass to orbit than chemical propulsion. If one pays attention to the way that spin is generated, and particularly if it is externally excited by means of an electric motor for example, then performance levels can be optimised for a range of mission options. It was with this in mind that Ziegler and Cartmell [11] investigated the potential for motorised tethers in 2001. In this paper the three mechanical options for upper payload release were re-considered; for a hanging, swinging (librating), and spinning tether, including a form of the ‘7+4d’ rule of [51]. It is shown in [11] that such rules do not take all the possible dynamical effects into consideration. An extended rule is derived, for both orbit raising and lowering, which takes orbital radius, tether length, and angular orbital and tether pitch velocities into account. This is shown to work well in some practically useful data cases. A Motorised Momentum Exchange Tether (MMET) on a circular orbit is considered and the nonlinear ordinary differential equation for this is used to develop an analytical spin-up criterion, which can also be compared with appropriately interpreted results of numerical integration of the governing ODE. This paper shows that a motorised tether can improve on the orbit raising

performance of a librating tether by two orders of magnitude, and the librating tether is roughly twice as effective as its hanging counterpart. The paper also distinguishes between performance and efficiency and suggests that in some circumstances short tethers can be advantageous in terms of efficiency (orbit raising parameter divided by tether propulsion length) despite their lower actual performance (measured in terms of the orbit raising parameter) than for longer tethers. Tether retrieval is the opposite of deployment and is equally important in dynamical terms. Retrieval of a subsatellite to a larger vehicle, specifically a space station, is examined by Djebli et al. [62]. This work was published in 2002 and concentrates on laws for retrieval (and also deployment) specifically combining ‘simple’ (linear or exponential retrieval) and ‘fast’ laws in which specific acceleration profiles are proposed. This would be applicable to passive momentum exchange tethers, MMETs, and potentially to ED tethers too. Fast (hyperbolic) retrieval is particularly advantageous because it tends to damp transverse vibrations in the tether, particularly when preceded by a simple sinusoidal retrieval law. Although ED tethers are dealt with in Section 3 it is pertinent to mention at this stage the work of Pela´ez et al. of 2002 [63] in which an ED tether is modelled as a two-bar system. The bars are articulated so that they can rotate relative to each other by means of an assumed universal joint. One end of the two-link tether is attached to a massive host spacecraft and the other end comprises a point end mass. The dynamics of the tether are modelled analytically by means of Lagrangian dynamics, and the ED forces are introduced within the generalised forces terms since they do not derive physically from a potential. It is important also to note that this model is adapted to the ProSEDS mission, [16,49], and is based on the intention that for such systems the second section is non-conductive and gravity gradient stabilised, thereby simplifying the general equations of motion. The authors consider this work to be an evolution of the dynamics of a dumb-bell model, applicable to ED and purely mechanical tethers. It shows that two modes of motion in the form of libration and lateral oscillation are both unstable in the absence of damping, the former growing slowly whilst the latter develops more quickly. The lateral instability is shown to grow particularly rapidly above a critical value of the ED to gravity gradient force ratio. An in-depth treatment of the rigid body dynamics of tethers in space is given by Ziegler [54]. In this work the dumb-bell tether is modelled at various levels of accuracy, and approximate analytical solutions are obtained by means of the method of multiple scales for periodic solutions. Comprehensive dynamical systems analyses are summarised for different configurations and models, and global stability criteria for a rigid body dumb-bell tether, in both passive and motorised forms, are defined and investigated. Further treatment of the spin-up criterion of [11] is also provided. Further work by Mazzoleni and Hoffman, [58], is of interest and relevance here, dealing

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with the end-body dynamics of artificial gravity generating, elastically tethered, satellite systems, undergoing nonplanar spin-up. In 2007, McKenzie [13] explored the dynamics of the MMET to include an analysis of the system on an inclined orbit and while undergoing deployment. Similar modelling techniques were used to investigate space-web dynamics, producing a stability map of the rotating space-web. Further relevant cross-references are available in [2,14,30]. 2.4. Tether models in which flexural effects are introduced The implication of the general dumb-bell model approach is that the tether is treated as a rigid body, however, that is not necessarily the case if axial stretch is allowed, and although then it may still be a dumb-bell in appearance it is no longer a rigid body in mathematical terms. Therefore, once that freedom is accommodated it becomes interesting to cater for further, more generic, forms of elasticity within the tether. The rigid dumb-bell tether is useful, however, not only for gaining an understanding of general global motions of a tether in space, but also as a fundamental tool for mission conceptualisation. In practice it is almost certainly the case that elastic models will be needed, and particularly so when very high accuracies are required both in predicting the tether location and orientation, but also in properly understanding the deformation of a tether in cases where the application is particularly demanding. This will frequently be the case in high-performance multi-line systems with high levels of built-in fail-safe redundancy, and also in tether-based structures such as orbiting stellar interferometers and space webs. In the case of the orbiting stellar interferometer DeCou [64] showed in 1989 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 into straight lines due to the tension created along their length by the corner masses as the whole system rotates. The other case is where the corner masses are zero and the tether mass density is finite, and then the triangle will necessarily deform into a circle. In practice we get something in between and this is shown in the paper by including finite corner and tether masses, along with in-plane deformation of the tethers. The tethers are broken down into segments and an iterative procedure is used to calculate the static shape that the system assumes. Axial stretch has already been mentioned as a parameter of fundamental importance in real systems. In long, high-performance propulsion tethers this could be very considerable, and will severely limit the applicability of rigid body modelling to anything other than system conceptualisation. A straightforward but compelling piece of work by Bernelli-Zazzera from 2001, and reported in [9], in which motion control is proposed for a tether in which

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stretch is allowed, showed that effective control can be applied by means of a short boom to which the deployer end of the tether is attached. The tether is free to move essentially as a conical elastic pendulum, and a terrestrial example is given in which a gravitational restoring force is included in a vertically orientated system. The stretch of the tether results in ‘bobbing’ oscillations which couple with the pendulum motion. The work is also of interest from a control and instrumentation point of view. The control is by means of planar rotational motion of the boom and is used within a linearised system model, noting that simulations were performed using a nonlinear model. Feedback of tether strain rate is used, in practice by means of an accelerometer within the end mass, and maximum damping of the bobbing oscillation is used to evaluate the optimal gain for the system. It is shown that the out-ofplane angle of the tether decouples from the rest of the system dynamics and so is uncontrollable, thus reducing the system to a planar elastic pendulum. The relevance of the work is in the effectiveness of the boom control actuator in minimising both the bobbing and the planar swing oscillations using simple actuation and control. In contrast with this Hokamoto et al. [65] consider a tethered ‘space robot’, conceptualised in their 2001 paper as a rigid body at the end of an elastic tether connected at its other end to a larger rigid body in the form of a main satellite. The system is constrained to motion in the orbital plane. Two links are attached to the robot satellite and are used as a manipulator to control the system. Torques are applied momentarily to the manipulator to effect the control and although strain energy in both axial and lateral tether deformation is initially included the paper does not state if and how this is developed, although the theoretical principles are all summarised. Extremely good tether swing minimisation is achieved for pragmatic data. Tethered interferometers based on different constellational configurations are considered by Quadrelli [66], with a general analysis for an n-body system where each body represents a spacecraft within the three-dimensional constellation and each interconnecting tether comprises N point masses, connected by massless springs and viscous dampers. Quadrelli points out that there are two ways to treat tether deployment with a lumped mass model; either the mass of the mass-points is kept constant, with varying number which is said to imply a mass creation and elimination procedure, or the number of mass-points is kept constant so that their masses will vary. Clearly, the first approach is more complicated than the second and so a general analysis is given for that, with a three spacecraft model used as a basis for the derivation, as briefly summarised in the paper. An important feature of this paper is that thermomechanical modelling is included, and that a link is provided between the dynamics and control aspects of tethered formation flying. This is specifically for interferometer applications which concentrate on a threespacecraft/two-tether system and a four-spacecraft/threetether system; noting back to the work of McKenzie and

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Cartmell [12] and McKenzie [13]. The ultimate aim of such research is to generate reconfigurable control schemes for very general configurations of tethered interferometers. The paper by Bombardelli et al. [67] investigates the issue of rotation plane change of a dipole-like interferometer comprising two end-located collectors and one central combiner, and proposes an open-loop control strategy for high-precision re-targetting. An interesting concept for a tethered space manipulator is given by Woo and Misra [68] in which a tether is proposed as a means of extending the range of a manipulator with little mass and fuel cost. Their system comprises a two-link manipulator, a tether, and a spacecraft. Modelling assumptions are that the robot manipulator’s arms are rigid links and the tether is rigid and straight, with a point mass located at each joint in the system and at the end effector. The centre of mass of the system is coincident with the centre of mass of the spacecraft, and describes a circular orbit around Earth. Planar motion of the system is assumed, and four angular generalised coordinates are used to define this. Although the tether is considered to be in tension, and rigid as a result, the authors concede that there can be theoretical conditions in which the tether tension goes negative, and therefore changes to compression. Their remedy for this is to calculate tension from an analytical expression during the simulations in order to ensure that the results are physically meaningful. Time histories of this, together with joint torques and overall end-effector position, are calculated and feasible paths for the end effector are given with certain torque restriction and various initial conditions. The end result of this is an algorithm, based on a globally convergent form of Newton’s method, for determining whether a feasible path exists between the end effector at its extended vertical position and the desired end point. Ishige et al. [69] introduce the concept of using an ED tether for space debris removal, on the basis that it can be considered initially as a simple dumb-bell (assumed to be rigid) but with significant tether mass lumped into a single point, and then as a discretised mass model within which flexibility is allowed by using parallel springs and viscous dampers as interconnections between the discrete tether masses. The geomagnetic field is modelled as a single magnetic dipole with titled axis and the magnetic field vector expressed in inertial coordinates. The tether debris removal strategy is based on a sequence of tether deployment and ED activity by which means the debris item is lowered sufficiently to burn up in the atmosphere. The tether flexibility couples with the orbital dynamics, although the paper does not discuss this in particular detail, however, tension variation effects are noted. Practical guidelines are given for the application of such a system, notwithstanding that the mass of the debris item, compared with that of the service satellite at the other end of the tether, could significantly affect system stability. Some useful further references relating to this work are in [62,63].

2.5. Control strategies and models In this section the emphasis is on proposals and methodologies for the application of some form of control of tethers in space. A highly readable and useful discussion of the use of motor driven momentum exchange tethers for lunar and interplanetary exploration is given by Puig-Suari et al. [70] and is another exposition of the motorised concept also proposed by Cartmell [53] and also cited in Refs. [4,10,11,13,24,27,54,55,56]. In [70], the authors explore the use of uniform tethers and tapered tethers, with a discussion of the attendant tether mass ratio. The important conclusion is made that tethers can be superior in mass terms to traditional chemical rockets for low-speed manoeuvres, but inferior for high-speed applications. Useful relationships between the energy required to spin up a tapered tether and energy converted by solar cells per unit area are given, within the approximate range 0.15–1.5 m2/kg of payload. The paper also summarises the problem of the provision of counter-inertia with motor drive designs. This is also a feature of the work of [53] and its extensions cited above. The authors of [70] also state that orbiting tether facilities can be extremely efficient in missions where many spacecraft launches are proposed, with particular advantages of general simplicity and reusability of the tether systems, however, it is generally acknowledged that this does not necessarily extend to the orbital control and maintenance requirements for tethers which can be rather demanding, particularly for high-performance interplanetary propulsion [13,53,54], in addition to ancillary hardware reliability problems, particularly in storage and deployer systems, and brakes. Several hardware and software implementation studies have taken place, with an early contribution provided by Gwaltney and Greene [71] in which the ‘Getaway Tether Experiment’, GATE, is discussed. A simple dynamical model of the planar libration dynamics is used in conjunction with a tether tension law based on length and length rate feedback, and other (cited) work has shown that this can produce desirable stabilisation characteristics during deployment and retrieval. In [71], the authors investigate the implementation of such control schemes within prototype hardware which is specifically designed for space flight; cross-reference with [23,72] for further discussions of specific hardware designs. The mechanical details of the implementation examined in [71] (stepper motor driven reel and wind mechanisms) are perhaps less important than the techniques used, for which simulations showed that the planar libration amplitude of an uncontrolled test tether of 10 m in length would reduce from 7.01 to 1.31 within 53 s, whereas yo-yo control, based on length variation at libration amplitude zero and peak excursion values, could get this down to 41 s, and phasing control, based on length variation at angular positions slightly before these points, could improve that further, with times down to 36 s. These predictions appear to have been backed up by experimental tests, although relatively little detail of these is given in the paper.

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An interesting control application for tethers is in the dynamic isolation of payloads from a mother satellite or spacecraft. Ohkami et al. [73] discuss this in the context of payload and space station interactions for microgravity control in the payload. A three mass system is modelled, comprising the base vehicle, the platform, and a ballast mass, connected in series by two tethers. Platform translations and rotations are considered and the equations are linearised on the assumption that for microgravity applications the deflections from the equilibrium state are small. Highly accurate microgravity manipulation is available for simple feedback control. In the case of large, controlled, tether motions, as would be required for waste disposal and capsule re-entry operations, nonlinear dynamics are unavoidable, but alternatives to fully analytical nonlinear controller design such as fuzzy logic can provide very effective control. This is discussed for a tether initiated re-entry application and a waste disposal system by Licata [74], in which a feedback system based on simple fuzzy logic rules controls the variable rate deployment for a deployer reel-brake assembly design, within a simulation, using realistic data. It is worth noting that this procedure is applied to medium length tethers of up to approximately 25 km. The control of robots remote from their space vehicle is revisited again in the work of Nohmi et al. [75]; also refer to [68], where translational momentum of the centre of mass of the tethered robot is controlled by tether tension, and angular momentum control, with respect to the tethered robot’s mass centre, is based on proper control of the tether attachment point and tether. These controls are effected by manipulations of tether tension and link motion in the case of a generalised robot in the form of n+1 rigid bodies (links) connected through rotational joints. The tether itself is considered to be a massless rigid link from the centre of mass of the spacecraft, subjected only to tension and with a time dependent length. A reaction wheel, jet, or thrust, is required to control angular momentum about the tether. It is shown that the link motion of the tethered robot can be satisfactorily split into two sub-tasks i.e. end-effector motion and tether attachment point motion, and that compensation for significant impulsive disturbances is robust and effective. Tension moments for four short tethers used to connect two spacecraft halves have been shown by Kumar and Kumar [76] to be good for the control of two aspects of system motion by means of a combined open-loop control law together with a simple feedback scheme. The motions considered for control are longitudinal system drift with respect to the ground station and attitude excitation induced by eccentricity. This work relates to the TSS as mentioned in Section 2.2, on the assumption that the TSS comprises two identical satellites connected through very short tethers, with the anchor points located on the principal roll axis and symmetrically offset from the centre of mass of each satellite. Tether mass is neglected and the planar angular motion case is considered. The pitching angles of the two satellite halves, the tethers, and the tether

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length define the motions of interest and comprise three angular and one translational generalised coordinate. Length variation can be controlled in order to force the system into certain motions and ultimately a special hybrid tether length control law is proposed. It is shown that effective control can be achieved using tethers as short as 10 m. The combination of open-loop and feedback control, in this context, results in a significant improvement in attitude precision for system alignment along the line-ofsight. This is proposed as a viable alternative to station keeping manoeuvres required for geostationary satellites, particularly in cases when onboard fuel is nearing exhaustion. Offset control of a tethered sub-satellite from a large platform is investigated by Pradhan et al. [77], on the basis of the TSS concept once again, where the offset mechanism takes the form of a manipulator attached to the platform capable of providing movement of the platform end of the tether in the local horizontal and vertical directions. The tether is modelled as a flexible string and the assumed modes method is used for discretisation. Motions are restricted to the orbital plane, and generalised coordinates for platform and tether pitch, together with tether modal coordinates, are used to define the system motions. Damping is usefully included by means of Rayleigh’s dissipation function and the generalised force vector represents momentum gyros located near the centre of mass of the platform and thrusters at the tether subsatellite end. Modelling accuracy is determined by checking the total system energy and comparing the frequencies of the linearised system with those available in the literature. The Feedback Linearization Technique (FLT) is used to control the attitude dynamics whereas a robust LQG control is used for the vibrational modes. The paper concentrates on controller design and overall offset control is seen to be effective for the regulation of platform pitch and tether vibrations but less so for tether attitude, for which large offset motions are required. A HypersonicAirplane Space-Tether Orbital Launch Vehicle (HASTOL) architecture has been proposed by Hoyt [78] in which several sub-concepts are proposed within the architecture for the transportation of large payloads into Earth orbit. The concept specifies a hypersonic aeroplane to carry a substantial payload up to an altitude of 80–100 km at a speed of Mach 10–13. The aeroplane is intended to rendezvous with the tip of a long rotating tether which swings down from a massive facility in Earth orbit. A grapple vehicle at the tether tip receives the payload which is then pulled up by the tether into orbit. The system is said to offer launch cost reduction because a conventional launch vehicle requires a total DV of around 7.5 km/s whereas the HASTOL launch vehicle only needs to provide about 3.5 km/s to the payload. This paper summarises concepts for three different tether systems for use within HASTOL and shows that a rotating tether is optimal, particularly if it connects with the highest possible apogee for the hypersonic spaceplane. Methods are given for

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maximising the rendezvous window, although this is still measured in seconds. The work confirms, entirely by means of numerical simulations, that the HASTOL concept, as defined, is pragmatic and controllable. Fujii et al. [79] consider a three degree of freedom nonlinear analytical model for a terrestrial deployment model for a tether with a floating balloon at its upper end, in order to obtain an understanding of the rotation, attitude, and deployed length when the simple balloon tensioned system is subjected to aerodynamic drag and is under the influence of a specific control law. The control law is designed to suppress the motion of the system and to adjust the tether length. The tension supplied by the balloon is of the order of a few Newtons and experiment and simulations based on the analytical model are compared. The concept of virtual mass is incorporated into the analytical model’s equations of motions, whereby it is found that the fluid surrounding a body which is accelerating within it (like the balloon in this case) seems to increase the mass of the body, and this is not negligible when the orders of magnitude of the mass of the body and the virtual mass are close. In the case of this system it is found that the presence of virtual mass improves the accuracy of the model. Whilst this in itself is not directly relevant to the performance of tethers in space it provides a very good means of testing important microgravity effects, with the assurance of a fair degree of controllability. Attitude stabilisation of tethered spacecraft confirmed as a major issue and configuration-based control of attitude has been shown, by Kumar and Yasaka [80], to be an elegant solution. In this work the authors start from the general literature finding that a single tether connecting a main satellite or vehicle to an auxiliary mass requires feedback control to ensure attitude stability of the satellite, but that a two-tether system can improve on this performance. On this basis these authors present a kitelike configuration comprising three tether spans, the top two emanating from points on the (upper) satellite symmetrically offset from the centre of mass and terminating at a common connecting point below from which the third span hangs down, ending in the auxiliary mass. The authors summarise a nonlinear non-dimensionalised Lagrangian model comprising 10 generalised coordinates and show by means of a stability analysis for the linearised system about equilibrium that certain physical constraints are necessary for stability, but that this is achievable and potentially inherently so; see also Quadrelli [66]. The single tether stabilisation problem is discussed by Cho and McClamroch [81] and the control objective here is slightly more strict, requiring not only attitude control of the satellite but that this is consistent with small tether motions. They do this in two ways, initially by decoupling the attitude dynamics from the tether dynamics and then designing in a linear feedback to stabilise the attitude, and also by using a Kalman decomposition to decouple uncontrollable modes and then using linear feedback to stabilise the controllable modes. They conclude that for roll-yaw attitude stabilisation, which is more demanding

than pitch control, the Kalman decomposition approach works best because less actuator movement is required and the tether dynamics are generally less affected. We return to ED tethers, and a model leading to a specific type of dynamic instability when working in inclined orbits in the paper by Pela´ez and Lara [82]. The instability is independent of tether flexibility and so the tether is modelled as a dumb-bell with end masses. The geomagnetic field is represented by a non-tilted dipole model and constant tether current is assumed. The electrodynamics force the system dynamics equations and the paper gives a full account of the stability of the tether in terms of the influence of the orbit inclination and a parameter representing the magnitude of the ED force on the tether. A numerical algorithm based on the Poincare´ method of continuation of periodic orbits is used to extend previous asymptotic analyses. The paper shows that high inclinations are not initially seen to be appropriate for vertical ED tethers and it is shown that for a given inclination there is a critical value of the ED magnitude parameter beyond which destabilisation is significantly accentuated. Tether current control can help to alleviate such effects, but it is recommended that such tethers are generally better off being operated away from this sort of threshold. The paper also shows that there are many unstable periodic solutions to this tether system and that such regimes are unsuitable for long-term ED tether operation. Introducing motor drive to a momentum exchange tether has many advantages but it has been shown by the authors, and others [11,13,26,53,54] that the interactions between local motor drive leading to tether spin and orbital mechanics are far from straightforward and that such systems are capable of very rich dynamics. Cartmell et al. [4] show that scale modelling, and performance prediction based on this, is a useful way forward when attempting to generate pragmatic data for the dynamics of controllable motorised tethers. In [4], it is shown that a symmetrically designed motorised tether, with the motor drive placed centrally and driving the two sub-spans, with inertial counter-balance, provides a basis for optimised performance. The proposal utilises the concept of payload release symmetry whereby the two payloads are released simultaneously from the ends of the tether sub-spans at the point when the system is aligned normal to the tangent to the orbit. Therefore the inner payload is de-boosted and the outer payload is boosted. The paper treats a simplified terrestrial (on-ice) model by means of classical scaling theory using the Buckingham Pitheorem. It is shown that even for the restricted dynamics of this system that certain very important trade-offs between mass and geometrical parameters have a significant effect on the system’s ability to spin-up, and it is proposed that insights and enhancements at this level are likely to improve the performance of motorised tethers in orbit. The work also provides a basis for full size system generation from the scale model, or vice versa. Tether vibrations were also investigated, by means of a threedimensional stretched string model, and scaling laws

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applied to this model, and the on-ice rigid body model showed that there is a numerical incompatibility when trying to scale rigid body spin-ups and flexural tether vibrations at the same time. Therefore, a clear case for multi-scale modelling is made in this paper. A simple despin concept is also proposed for payloads in order to direct angular momentum in spin of the payload at and after release back to the spinning tether system, and an initial two degree of freedom nonlinear model is introduced and discussed. One of the questions raised by this work relates to the dynamics of tethers after payload release, and how they can be controlled when the tether effectively becomes a trailing structure. This general problem is rather universal and generally independent of tether type or the configuration in which it is used. The paper by Rossi et al. [83] of 2004 provides an interesting account of the likely periodic motions of a tether trailing satellite, with attention paid both to the motion of the satellite and the tether. The scenario that the paper considers is when a tether connecting two satellites is cut as a consequence of an accident or a planned manoeuvre. It is assumed that the Earth centred frame is inertial, that the satellite can be modelled as a point mass, the tether is homogeneous with uniform density, the torsional and transversal vibrations of the tether can be neglected, and elasticity follows Hooke’s law. The model comprises partial and ordinary differential equations and applies the well-known wave equation in an orbital context, together with the effects of atmospheric drag and Earth oblateness. This significant work shows that the existence of periodic solutions for such a system does not depend on the equilibrium state when gravitational and oblateness terms predominantly drive the dynamics. The important features relate solely to tether density, length, flexibility, and rotational speed. However, shorter stiffer systems tend to exhibit periodic motions about their equilibrium states. In the case where atmospheric drag influences the system at all significantly it is found that tether trailing satellites are strongly influenced by the equilibrium state; this is because the gravitational and oblateness forces are uniformly bounded independent of position, whereas drag forces do not behave like this and so linearisation about the equilibrium states is required. So, the existence of periodic motions with bounded forces is found to depend just on tether parameters, but unbounded (linear) growth depends on the equilibrium states. Several interesting cross-references can be found in [9,42,57,62,65, 66,68]. 2.6. Practical tether designs and proposed system technologies It has already been shown that an important objective of tether modelling is to generate data that can be used for pragmatic designs which will perform optimally and predictably when in orbit. Carroll proposed a preliminary design for a 1 km/s tether transport system [84] in orbit around the Earth, the Moon, or Mars. This paper is

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interesting because it considers a single-ended (single payload) tether in what the author defines as a ‘barely spinning’ mode, where the system spin rate is approximately synchronous with the orbital period. The author gives substantial numerical data for practically useful mission applications about Earth, the Moon, and Mars, on the basis of how tethers might operate within certain payload delivery and retrieval scenarios about those bodies. In addition the paper discusses practical proposals for a ‘traction-winch’ tether spool-store system with controllable feed and cites interesting critical reeling rates for different stored lengths up to 200 km. The paper also discusses concepts for testing thick tethers under highrepetition cyclical reeling at low temperatures, and capture hardware design is proposed based on the criteria of large capture zone and design simplicity, in the form of a hook and bag system. The paper also considers capture and release transients that inevitably arise and which drive sudden changes in equilibrium tether tension and length, together with some comments on general deployment strategy and general operational conditions. Retrieval of a tether and point-mass payload to a massive spacecraft is considered in some detail in the paper by Chernous’ko [85]. The configuration comprises a reeled out tether and endmass payload swinging from below the spacecraft and is restricted to a planar analysis. The forces acting on the system comprise tether tension, gravitational force, and inertial forces (centripetal (centrifugal) and Coriolis). The author suggests four different practical ways of controlling the retrieval process, firstly by using small motors on the satellite to generate reaction forces perpendicular to the tether, moving the tether emanation point on the spacecraft relative to the spacecraft in order to suppress oscillations, controlling small deviatory motions of the spacecraft in order to suppress oscillations, or controlling tether tension during retrieval. This paper considers several important cases, specifically for constant tether length, oscillations at a constant winding rate, small tether oscillations, nonlinear oscillations, and control of the retrieval process. Control is achieved by a two-stage process in which the tether is initially maintained at constant length during which phase trajectories are used to determine the necessary constant rate of winding in the second stage. The second stage proceeds and operates at the calculated winding rate until retrieval is complete. Trade-offs between tether mass, strength, and longevity are of fundamental importance and in the case of momentum exchange tethers one of the most interesting concept proposals to emerge has been that of the Hoytether, Forward and Hoyt [41,86], in which a multiyear lifetime is proposed for an open tubular tri-axial net. This design consists of axial load bearing primary lines with cross-linking at intervals by diagonal secondary lines, which are only loaded if the section of primary line that they surround fails for some reason. The most likely cause of failure in such a scenario is due to high-velocity micrometeoroids or space debris. Due to this design the

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effect of the damage is localised to the failure region and the load is re-distributed within the secondary lines around this region. The authors claim a Hoytether lifetime in the order of decades rather than the less-than-full-mission capability of single line tethers. Further practical ideas resulted in the Rapunzel small tether mission for tether assisted payload re-entry. This is discussed in the paper by Sabath et al. [87] and was intended to be a small tether mission whose deployer was tested in a parabolic flight in 1995. The deployer consists of a tether box containing a spooled tether of 63 km and a brake and compensator based on textile industry technology. The deployer was found to perform quite well during the microgravity conditions of the parabolic flight with good tension control in evidence. However, despite the effectiveness of tether designs and their deployment systems the space debris mitigation problem is one that has had to be addressed in recent times. The work of Rex [88] highlights some of the spacecraft design changes that would be effective in debris mitigation. Two principal approaches are highlighted; passivation, in which onboard stored chemical propellants are removed in order to prevent debris generation by explosions, and the deliberate de-orbiting of larger orbiting objects to reduce the possibility of collision. ED tethers are suggested for de-orbiting applications, and this has also been explored by others, notably Hoyt and Forward [86]. Clearly the orbital debris problem also affects the use of tethers and this is explored further in the work of Draper [24]. Very short tethers could be feasibly used to achieve controllable satellite pitch and roll attitude manoeuvres, Kumar and Kumar [89], and an in-depth dynamical model description is given for a short four-tether system connecting a satellite to an auxilliary mass, and the synthesis of open-loop tether length control laws. Good control of complex manoeverability is obtained for sufficiently slow tether length variation, together with small amplitude oscillations about the desired final equilibrium position. This could be an important methodology for safe satellite operation in certain circumstances and the general issue of safe operation of tethers, particularly in unscheduled operations, from the ISS is discussed in a practically focused analysis by Trivailo et al. [90]. Unscheduled operations are meant to cover instances of unexpected severance, and interference between the tether and other hardware. Both type of event has been shown to be possible in tether retrieval operations, particularly if socalled skip-rope modes are initiated. The paper shows that instabilities can be caused by an excessive retrieval rate and also by skip-rope motion, both of which can give rise to severance or interference with other hardware. Dynamic simulations show that interference with the ISS itself would be likely, with severance as the final outcome. Tether missions involving interplanetary propulsion or the orbit raising of major payloads will inevitably require the use of a reusable space plane system capable of liaising with a tether for payload handover. Considerable conceptual work on this issue has been carried out, reported by

Hoyt [78] and Grant et al. [91], and, as already mentioned in Section 2.5, has resulted in proposals for the Hypersonic Airplane Space Tether Orbital Launch (HASTOL) vehicle. This technology overview provides insights into the possibilities of flying a 15 tonne payload in a ballistic arc to reach Mach 10–13 at an altitude of 80–100 km.This liaises with a grapple mechanism at the end of a rotating 600 km tapered tether in a 700 km orbit, as a ‘highway to space’. The authors promote HASTOL as a completely reusable, cost-cutting technology for Earth-to-orbit space access. In a similar vein Hoyt [92] discusses the design and simulation of a tether boost facility for transport from LEO to GTO. Proposals for boosting 2.5 tonnes from LEO to GTO every 30 days are discussed in the paper and it is also stated that the same facility could be used to boost 1 tonne payloads to LTO. The tether in this system is tapering but comprises multiple lines to provide both strength and redundancy, possibly in the form of a Hoytether [41]. The orbital dynamics are summarised and the use of an ED tether is discussed. The theme of deployability continues to receive attention and Pascal et al. [93] have shown that the use of a crawler sub-satellite which moves along the tether during retrieval can be stabilising, particularly if combined with appropriate length control laws in the form of an intermediate scheme generalising the previously proposed conventional scheme and the crawler scheme. The paper presents an in-depth dynamical treatment of such a scenario and shows by means of a numerical simulation that the so-called intermediate scheme reduces the amplitude of oscillation during retrieval several times over those of the conventional or crawler schemes. The use of crawlers is also examined in the paper by Goff and Siegel [94] in which two massive nuclear electric crawlers are fitted to the sub-spans of a symmetrical momentum exchange tether with centralised facility. This is described as boot-strapping and is proposed as a means of angular speed control in which the two crawlers move in or out from the centre as required. Control and stabilisation of tethered systems is examined for three body configurations by Misra [95] in which a double-ended payload system, complete with sizeable centralised facility mass, is analysed in detail. The assumptions made relate to inextensible, straight-line, mass-less tethers, with point mass payloads, with the system COM on a circular orbit and undergoing planar dynamics. Local angular coordinates are introduced, allowing the two sub-spans to be at different angles, to give a two degree of freedom model for which four principal equilibrium conditions are evaluated. The stability of the equilibrium conditions is investigated for small perturbations and the eigenvalues of the characteristic equation show that at best there can be marginal stability for certain specific conditions, but no asymptotic stability. Spinning tethers are normally assumed to operate as dumbbells, with axially aligned sub-spans due to centripetal stiffening during rotation. However, Misra [95] shows that in cases where this is absent for some reason then the

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system sub-spans may well take other orientations, and in such cases practical issues of alternative stability have to be considered. This could have relevance to both pre and post tether spin-up scenarios. Powell et al. [96] provide interesting insights into the use of technology for magnetically inflated cables for the construction or deployment of large and highly rigid space structures. The idea is based on launching the magnetically inflatable cables (MIC) as a compact package of coiled superconducting cables, which would be cryogenically cooled and electrically energised on reaching orbit. This would result in magnetic repulsion, which would then allow the coiled package to self-deploy into the intended structural configuration. A network of high-strength tethers would be required to hold the superconducting cables in place. This has some similarities in terms of the end result with the work on space webs by McKenzie and Cartmell [12] and McKenzie [13]. It is possible that long structures could also be made this way, which could themselves operate as large capacity tethers. Hybrid designs involving two or more tether technologies could be of great utility for certain space missions. A case in point is the MXER concept [19,21,22] as discussed already in Sections 2.1 and 2.2. In [22], Sorensen considers the conceptual design of an MXER tether boost station, concluding that a single tether in an elliptical equatorial orbit could replace staged tethers, using propellantless ED reboost with highly error-tolerant payload catch mechanisms and tether-end mass concentration. An interesting practical problem associated with tether system operation in space relates to the detection of tethered satellites or payloads, as distinct from free flyers. This problem is addressed by Choe et al. [97]. The ideas in this paper are based on the fact that the constrained motion of two or more tethered satellites is quite different to that of a single satellite with the same position and velocity at some given instant. The effect of this is that a conventional orbit detection algorithm for a single satellite will over predict the orbit of the upper satellite in a tethered pair and, conversely, will under predict the orbit of the lower satellite. Two and three satellite tethered systems are modelled using a multi-body formulation in which constant length constraints are modelled by means of Lagrange multipliers. The paper shows that the constraint force between two satellites in a tethered system, due to the tether and gravity, can be regarded as a tension expressed by a Lagrange multiplier which can be used to give the tether acceleration, and which can be extended to an nbody system where some of the bodies are tethered and some are not. Observations of the motions of all possible combinatory pairs of satellites within the system allows tethered and untethered satellites to be distinguished from one another, by recognition of non-zero tether acceleration per unit length (Lagrange multiplier/mass of each satellite). The information provided by this sort of technique will be of increasing importance as tether missions increase in quantity in the future.

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Control of tether motions has been a recurring theme throughout the paper, and for good reason as robust assurance of controllability is necessarily a central theme in all architecture and mission plans. The work of Williams et al. [98] has shown that so-called receding horizon control using quasi-linearisation and Chebyshev pseudospectral approximations can effectively be applied for tethers, and that this does not require extensive computations, and in fact is reducible to the solution of simultaneous linear equations. This has obvious implications for practical system implementation, with good disturbance rejection and trajectory tracking capabilities. We conclude this section by referring to five papers in which specific missions for tethers are summarised. The paper by Jokic and Longuski [99] discusses the possibility of artificial gravity provision and free-return aborts for Mars missions, and proposes a massive tether system with a habitation at one end and a counter mass at the other, with rotation of the system about the COM providing Earth gravity-like acceleration within the habitation. The concept is centred around the idea of a propellant-free return of astronauts to Earth in the event of an aborted landing on Mars for 2014, 2018, 2020, and 2026 based on the NASA Design Reference Mission. Williams et al. [100] explain their concept for momentum-enhanced gravity assist of a spacecraft at a destination planet by deploying a payload on a tether from the spacecraft in such a way that it is boosted onto a new escape trajectory. Numerical simulations are used to validate the proposals made. A novel method for orbital transfer of a payload by means of a tether is offered by Kumar et al. [101] based on controlled deployment and retrieval of the tether. They confirm that short tether lengths are associated with higher-performance indices than systems with longer tethers (see Ziegler and Cartmell [11]). The Icarus student satellite project, Goldberg and Gilchrist [102], is a small active end-mass satellite developed at the University of Michigan for the ProSEDS ED mission. The roˆle of ICARUS is as a data collector and transmitter for tether deployment and dynamics, using GPS and an aspect magnetometer. Anselmo and Pardini [103] consider the survivability of space tether systems in orbit specifically around Earth and confirm that single line designs are unlikely to complete their missions in such a demanding environment, whereas a relatively simple knotted and looped design offers much better survivability of between 95% and 99%. A lot of related information is also available in [2,4,9,48,50,52,64–66,70,71,78,80]. 2.7. Deployment scenarios and mission plans Grassi and Cosmo [104] provide an investigation of the attitude dynamics of the SEDS system in their paper of 1995. The equations of motion for a rigid body SEDS payload deployed to a Delta second stage are derived assuming that the payload’s centre of mass is on a circular orbit, the tether is straight and inelastic, the gravitational potential can be linearised, and the attitude angles are

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small. It is shown that the pitch equation is decoupled from the roll and yaw equations and for simplicity it is argued that the remaining roll-yaw coupling can be neglected, as is the yaw damping term. Application of the Routh-Hurwitz stability criterion provides conditions for roll-yaw, and pitch stability conditions. The overall finding of the work is that payload stability for the SEDS concept is strongly affected by the initial conditions of the deployment, and that tumbling around the roll and yaw axes starts as soon as the payload is ejected from the Delta second stage. An attachment point displacement system, with motion of the attachment point along the negative pitch axis direction, is introduced and shown to be effective in reducing the tumbling effect about the roll and yaw axes. These findings are likely to be of general relevance to many different tether mission deployment scenarios. A general proposal for Earth–Mars transportation using tethers has been given by Forward and Nordley [61], in which rotating tethers in highly elliptical orbits operate about each planet. The paper starts with the observation that Hoyt and Forward [3] designed a three tether Earth–Moon system for rocketfree transportation and that much of the propulsion gain from this concept came from the outer EEO tether, principally because of the important and general rule that boost of any sort is best made from deep inside a planetary gravity well. The Mars–Earth Interplanetary Tether Transport System, or MERITT, uses the fact that a single EEO tether, in a highly elliptical orbit, can theoretically propel a payload to Mars. The tethers at each end are known as EarthWhip and MarsWhip, respectively, and rotate rapidly in highly elliptical orbits. The EarthWhip tether collects the payload from the delivery vehicle from Earth and releases it later when the tether is near perigee again and the sub-span is at the ‘high point of its swing’, this necessarily being when the tether is orientated normal to the tangent to the orbit at perigee. The payload gains velocity and potential energy from the tether and it then has sufficient energy to go on a high-speed trajectory to Mars, with no further boost needed other than mid-course correction. The payload is then caught at periapsis by the MarsWhip tether at the highest point of its rotation where it has its greatest velocity with respect to Mars, and is later released when it’s again at periapsis and at the lowest point of its rotation, with delivery to the Martian atmosphere. Importantly, the authors state that the system works in both directions and conclude with the significant proposal that the MERITT concept could be applied to other planets and moons, as a general Rapid Interplanetary Tether Transport (RITT) system. More unusual mission possibilities should also be mentioned, starting initially with the work of Maccone [105] who discusses the use of tethers to obtain magnified radio pictures of the galactic centre from distances of 550 AU. This uses antennas tethered to a spacecraft, with the whole system moving at uniform speed away form the Sun on a purely radial trajectory. Further work by Maccone [106], related to the Search for Extraterrestrial Intelligence (SETI), proposes

the use of two antennas, tethered together in two different ways, for the purposes of setting up a SETI receiver system inside the Saha crater on the far side of the Moon, in the absence of radio frequency interference. In both variants the tether doubles as the cable connecting the two antennas. The YES2 mission was intended to use a 30 km tether as part of a sample return system from a Foton-M3 carrier vehicle, where an inherently safe re-entry vehicle is returned to inhabited regions of western Europe, areas which would normally be well out of bounds for space return operations [72]. The inherent safety feature comes from the use of a low mass inflated return vehicle (of 5–15 kg), which reenters the atmosphere relatively slowly and at reasonable temperatures. The tether dynamics provide the correct reentry conditions for the re-entry vehicle, by means of simple length feedback control. Secondary applications for tethers are envisaged by Accettura et al. [107] in their paper on integrated propulsion missions to Mars in which they suggest that nuclear and superconductive magneto-plasmadynamic (MPD) propulsion could be combined for Earth–Mars trajectories, and that tethers could be used for artificial gravity during interplanetary flight, and as a space elevator in Mars Stationary Orbit, MSO. Considerable cross-referencing is available in the papers of [16,17,21,32,48,49,51,60,62,78,79,91,102]. 3. Electrodynamic tethers 3.1. Summary of operating principles The works of Johnson et al. [16] and Sorensen [18,19] have already been cited and discussed and it is worth reemphasising their importance and relevance in the applications of ED tether systems. The role of ED tethers is briefly summarised within the road map proposals made in [27]. Conventional ED tether applications harness the effect of planetary magnetic fields interacting with currents actively driven, or passively induced, in the tether. Somenzi et al. [108] investigate the stability analysis of a conventional but flexible ED tether in terms of libration and also lateral oscillations expanded as normal modes. Simplifying assumptions are applied, in the form of a circular unperturbed orbit, the system centre of mass being located on the parent satellite, constant tether length, and considering the satellite and ballast as point masses. The system comprises a satellite vehicle at one end and a ballast mass at the other, and a Lorentz-type ED force exerted on the tether, operating either as a drag force or as a propulsive force, dependent on the direction of the current flow. The analysis considers the system dynamics with and without the ED force, and it is shown that a source of attitude instability comes from resonances occurring between the out-of-plane libration angle and the ED forces, for non-zero inclinations. There is coupling between libration and the lateral modes so this instability also affects the lateral modes, but far more so for the odd

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modes than the even modes. The use of electromagnetics within formation flying is worth mentioning here, particularly when used in conjunction with tethers to hold interferometric spacecraft arrays in formation without the use of on-board propellant. Sedwick and Schweighart [46] showed that complete control of all relative degrees of freedom within an array can be achieved by electromagnetic diploes and tethers, and that satellite systems can be spun-up and controlled in terms of attitude and position in this way. Returning to more conventional ED tether thinking, Yamaigiwa et al. [109] discuss the dynamics of an ED tether applied to deorbiting a space vehicle; note appropriate cross-references to the terminator tether [86] and ProSEDS [16,49]. Once again the Lorentz force is exploited and the mass of the vehicle to be de-orbited is assumed to be a very realistic 1000 kg, with a 50 kg ballast mass at the other end of the ED tether. The authors show that there is a maximum value of the eccentricity of the initial orbit from which an ED tether de-orbit can feasibly be achieved, and that the limit of the eccentricity can be controlled by the Lorentz force, especially in the case of a small ballast mass and a short tether. Additional useful reference information is available in [16,63,82]. 3.1.1. The TSS-1R mission One of the most well-known tether test experiments was the TSS-1R mission, [47,110], in which certain space plasma-ED processes were to be explored, in conjunction with the orbital mechanics of a gravity gradient stabilised system of two satellites linked by a long conducting tether. The mission is important because the tether EMF and current reached 3.5 kV and 1 A respectively, providing significant insight to viable current collection processes and the physics of high-voltage plasma shields. The TSS-1R mission showed that motion relative to the plasma affects current collection and has to be taken into account in the orbital dynamics. 3.2. Practical electrodynamic tether designs and proposed system technologies The efficient operation of ED tethers relies inherently on optimal exploitation of plasma physics, and the dynamic response of ionospheric plasmas is a feature of the paper by Zhou [111] in which a hybrid analytical-numerical method is proposed to understand the dynamic response of a 2D magnetoplasma to a time-dependent current source imposed across the magnetic field, with the result that ionospheric plasmas are seen to respond to current sources induced by a pulsed tether through the excitation of Whistler waves and the formation of an expanding local current loop induced by field-aligned plasma currents. The author suggests that the method can be extended into three dimensions and intimates that nonlinear phenomena and boundary effects could well be important. Shiah et al. [112] considered the three-dimensional simulation of current collection in space, specifically in LEO, and related their

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work to the TSS-1R mission, using a Super Particle Simulation (SUPS) model for the local orbital environment. The plasma’s transient and asymptotic response behaviour is investigated around a three-dimensional satellite of realistic geometry. The plasma dynamics are highly complex and time variant, and it is particularly noted that the resolution time of a typical experiment is much longer than the simulated timescale, inferring that the experiment may not catch the transient effect in space but emphasising that the transient effects are still very important. The authors also note that the TSS-1R mission, despite failing to deliver on principal objectives due to tether failure, still showed that the measured collected current was, in fact, around 2.5 times the theoretical prediction [112]. Gilchrist et al. [113] undertook chamber tests of simulated ED tethers of different geometries operating in a dense, high-speed plasma. This important paper considers cylindrical, flat-ribbon, sparse-ribbon, and mesh tether geometries and shows that the tape tether may be the best design for bare ED tether geometries. The mesh tether, with its desirable levels of built-in redundancy, did not perform as well as the tape, and it should be noted that end-effect uncertainties also entered into the design assessments. Computation of current flow in a bare moving tether formed the subject of the paper by Onishi et al. [114], in which a particle-in-cell (PIC) method was used to calculate electron current in a bare tether moving at orbital velocity in the ionosphere. The PIC method was found to agree with existing theoretical work for the quiescent unmagnetised case, and performed well in simulating the current collection. In the flowing case particle–field interactions associated with the fluctuating potential field were found to appear and to enhance the current collection for the tether, but this enhancement was not fully understood at the time of writing. The transmission line characteristics of ED tethers are modelled by Bile´n et al. [115], with the TSS mission as the backdrop. A voltage dependent sheath model was developed for ED tether transmission lines and was implemented within the SPICE circuit simulation program. This allows complete tethers to be modelled by including circuit representations of the tether endpoints, and includes the interactions that they have with the tether model itself. Large current enhancements were observed at frequencies resonant with the input reactance, but at the expense of high RF power. It was proposed that an experiment with controlled power and loss levels would be useful, in order to enable the direct measurement of current enhancements. Morris et al. [116] presented ideas for the use of field emitter array cathodes (FEACs) which consist of up to several thousand micron sized cathode/gate pairs printed onto a semiconductor wafer for cold field emission at relatively low voltages, providing arrays capable of A/cm2 level current densities, as would be required for tether use. Returning to the coupling between electrodynamics and dynamical phenomena, Ruiz et al. [117] provided a Lagrangian model for an elastic tether from which they could perform modal

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analysis in order to investigate the stability of an ED tether. The intention behind this work was to create a model that could be used to simulate the dynamics of an ED tether on inclined orbits, where the tether is found to be affected by a slow-growing ED instability. Damping is also included, and both controlled and uncontrolled models are discussed. It is found that the system is sensitive to damping and that this generally enhances the system controllability. The dynamic stability of ED tethers on inclined elliptical orbits is modelled by Pela´ez and Andre´s [118] who study the combined effects of orbital eccentricity and ED forces on the attitude dynamics of an ED tether. Tragesser and San [119] give an analytical account of orbit manoeuvring with ED tethers in which the general perturbation equations are used to develop a guidance algorithm for ED tethers. The algorithm presented here is capable of performing any LEO transfer given a sufficiently long time for the manoeuvre, noting that the trajectories generated are not necessarily optimal. Tahara et al. [120] provide a basic study of electron collection by a bare tether satellite and undertook ground experiments in which metallic tethers were exposed to a simulated LEO plasma flow, in the presence of a magnetic field. The general finding from this work was that the current collection characteristics of a bare tether in space strongly depend on the plasma velocity and the surrounding magnetic field strength. The stabilisation of ED tethers forms the topic of a highly readable report by Hoyt [121] in which pendulum librations, transverse wave oscillations, and skip-rope modes are all investigated and control laws proposed in the form of feedback algorithms. It is proposed that the dynamics of ED tethers can be stabilised by means of such algorithms during extended periods of operation. Supplementary reference information can be found in [22,69,92]. Korepanov and Dudkin [122] give a short exposition of the research potential for ED tethers in the lower ionospheric layers (F-layer and below), however, they conclude that long duration space experiments at that location are not likely to be successful due to the high risk of failure from small high-velocity particles, however, this comes from considering single-line tether designs as opposed to multiline redundant tethers, which would presumably fare much better. 4. Conclusions This paper has attempted to provide the interested reader with a reasonably broad background to the field of space tether research. As has been shown, this is an extremely active and vibrant research area internationally, with considerable contributions made to the literature in recent years. Clearly, space limitations limit the coverage of this review and the authors admit full responsibility for the choice of citations, and for any incompleteness of the review work that has been carried out as a result of this. The intention has been to provide the motivated student with a significant reference resource, notwithstanding the

very large number of papers and manuscripts that have been published over the whole spectrum of tether activities. The review paper covers both momentum exchange and ED applications, and sets theories against mission and technology development agendas as far as practically possible. It is shown that both momentum exchange and electrodynamics can, both separately and together, provide practical and workable propellantless propulsion, as well as offering various de-orbiting and re-entry functions. Acknowledgements The authors are indebted to the international research community for the ideas presented in the paper and the second author wishes to thank his students and colleagues for their invaluable support and their many major contributions. The EPSRC studentship awarded to David McKenzie is also acknowledged. Finally, we would like to acknowledge our gratitude to Dr. Robert L.Forward, Dr. William Berry, and Mr. Michael Eiden for their interest in, and support of, the tether research that initiated at the University of Edinburgh in 1996 and which we then transferred to the University of Glasgow in 1998. References [1] Beletsky VV, Levin EM. Dynamics of space tether systems. In: Advances in the astronautical sciences, vol. 83. San Diego: American Astronautical Society; 1993. [2] Lorenzini EC, Cosmo ML, Kaiser M, Bangham ME, Vonderwell DJ, Johnson L. Mission analysis of spinning systems for transfer from low orbits to geostationary. Harvard-Smithsonian Center for Astrophysics; 1999 [preprint series no 4803]. [3] Hoyt RP, Forward RL. Tether transport from sub-Earth orbit to the lunar surface y and back! In: Proceedings of the international space development conference, Orlando, May 1997. [4] Cartmell MP, Ziegler SW, Neill DS. On the performance prediction and scale modelling of a motorised momentum exchange propulsion tether. In: Proceedings of the STAIF2003 conference. University of New Mexico; 2003. [5] Bekey I. Tethers open new space options. Astronaut Aeronaut 1983;21(4):32–40. [6] Carroll JA. Tether applications in space transportation. Acta Astronaut 1986;13(4):165–74. [7] Moravec H. A nonsynchronous orbital skyhook. J Astronaut Sci 1977;25(4):307–22. [8] Forward RL. Tether transport from LEO to the lunar surface. In: Proceedings of the 27th AIAA/ASME/SAE/ASEE joint propulsion conference and exhibit. AIAA; 1991. p. 91–2322. [9] Bernelli-Zazzera F. Active control of tether satellites via boom rotation: a proof of concept experiment. In: Proceedings of THE AAS/AIAA space flight mechanics meeting, p. 1225–40 [AAS. 01192]. [10] McInnes CR, Cartmell MP. Propellantless propulsion. In: Modern astrodynamics, vol. 1. Elsevier Astrodynamics Series. Oxford, Amsterdam: Elsevier Academic Press; 2006. [11] Ziegler SW, Cartmell MP. Using motorised tethers for payload orbital transfer. J Spacecraft Rockets 2001;38(6):904–13. [12] McKenzie DJ, Cartmell MP. Large tether and web structures in space. In: Symposium on the mechanics of slender structures, University of Northampton & Institute of Physics, 28–29 September 2006. CD-ROM paper under authors’ names.

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