Libration Control of Bare Electrodynamic Tethers Considering Elastic–Thermal–Electrical Coupling

Abstract

The paper investigates the elastic, thermal, and electrical coupling effects on the dynamics and libration stability of flexible bare electrodynamic tethers in the end-of-mission satellite deorbit. A high-fidelity model is developed by considering transverse and longitudinal dynamics and libration dynamics of the tether and environmental effects with the latest models for atmospheric and plasma density, as well as Earth gravity and magnetic fields. The long-term orbital and libration dynamics of the tether are analyzed by a nodal position finite element method and symplectic time integration. Orbital motion limited theory is used to model the electron collection by bare electrodynamic tethers, whereas a Fowler-Nordheim equation is used for a Spindt array cathode. The thermal effect and its coupling with the dynamics of electrodynamic tethers are investigated parametrically. It is found that the thermal effect significantly affects the stability of electrodynamic tethers, which must be considered in the stability control. Two practical and effective electrical current on/off control strategies are developed based on the libration energy and libration angles. Although the libration energy control is found to be more stable and efficient than the libration angle control, the latter is more appealing for practical applications due to its simplicity and low computational effort. NomenclatureA t , A s = cross section or stress area of tether and projected drag area of satellite, m 2 B g = vector of Earth magnetic field strength in the global frame, T c m = specific heat of tether material, J · kg · K −1 d = tether diameter, m e E = eccentricity of electrodynamic tether orbital plane H 0 = Hamiltonian of tethered system in equilibrium configuration, J I t = electrical current at electron emitter device, A L, L = total and characteristic length of tether, m L e0 L e = unstretched length (instant length) of tether element, m M yaw , M in , M out = components of electrodynamic torque in system orbital frame, N · m n orb = orbital mean motion, rad · s −1 Q = thermal flux, W · m 2 R e = electrical resistance in tether, Ω R, S, W = unit vector of orbital frame of tethered system T, T 0 = instant and initial temperature, K t = time t, n, b = unit vectors of the local frame of tether element V cc = potential bias at the cathodic end C with respect to the ambient plasma, V X, Y, Z = components of position vector in global geocentric inertial frame, m Z T = impedance of emitter device, Ω α, _ α = pitch angle and pitch angular velocity of virtual tether, rad α p , _ α p = periodic solution of pitch angle, rad; and angular velocity of virtual tether, rad · s −1 α s ∕ε = ratio of absorptivity to emissivity of the tether β, _ β = roll angle, rad; and angular velocity of virtual tether, rad · s −1 θ, θ 0 = north latitude and colatitude, rad λ = east longitude, rad μ g = gravitational constant of Earth, m 3 s −2 ρ a ,ρ t = atmospheric density and material density of tether, kg · m −3 σ = electrical conductivity of tether material, Ω −1...

Review of Control Methods and Strategies of Space Tether Satellites

“…A paper by Ganqiang Li developed two practical and effective electrical current with on and off control strategies on the basis of libration energy and librational angles. It is found out that the libration energy control is more stable and efficient [58]. The libration angle is more on practical applications because of its simplicity and low computational effort.…”

Debris Rotation Analysis During Tethered Towing for Active Debris Removal

“…In this analysis, the aerodynamic forces are omitted for simplicity. However, below the altitude of 500 km, the detailed analysis should be performed, since the differential aerodynamic forces can lead to the end bodies tumbling and the whole tethered system's oscillations [23].…”

“…Many concepts of ensuring the connection with the debris target have been investigated, including harpoons, nets, grapples and robotic arms [4][5][6][7]. The dynamics of the electrodynamic tethers and their interaction with the Earth's plasma and magnetic is the active field of research [1,23]. Towing the inactive target via a tether continuous to be a promising idea, and is being widely investigated [8][9][10][11][12][13][14][15][16].…”

“…Nevertheless, closed-loop control adds complexity to the system, requires careful consideration of sensor performance and may increase the control effort due to sensor noise. In [23], Li, Zhu, et al, analyse the dynamics of the electrodynamic tethers and show that the thermal effects affect the stability of the current-conducting tethers. The authors in [25] address the challenges of accurate and robust time integration methods applied to the space tethers dynamic models.…”

Libration control of bare electrodynamic tether for three-dimensional deployment

“…[22] that involved minimizing the error between the predicted libration motion of the tether and that in the previous period. Li et al have analyzed the thermal effect on the stability of an EDT and has proposed two practical electrical current on/off strategies for controlling libration energy and angles [23]. To improve the stability of the EDT system, Takeichi has proposed a current switching control strategy and a stability function with periodic solution to facilitate the control [24].…”