Spinning electrodynamic tethers (EDTs) are promising technologies for propellant-less orbital maneuvering and de-orbiting applications. The spinning configuration offers several advantages, such as flexibility to orient the EDT with respect to the local geomagnetic field, thus maximizing the Lorentz force, and the avoidance of the dynamic instability of EDTs aligned with the local vertical, which is due to continuous pumping of energy by the Lorentz force [1]. Additionally, under certain simplifying assumptions, the tether's spin plane that results in a Lorentz force that maximizes the variation of different orbital elements depends only on the orbital inclination. Therefore, the optimal tether's spin plane remains stationary for a given orbit [2] and for very fast-rotating tethers, the orientation of the spin plane fits this condition and remains constant over time [3]. However,  for a finite spin angular velocity (spin rate), the gravitational torque induces a precession of the spin plane and moves it away from the optimal orientation, decreasing the performance for the variation of the orbital elements. This work develops and studies a three-dimensional attitude model for spinning tethered systems affected only by the gravity gradient, with the goal of gaining insights into the dynamics of the spin plane, including the equilibria and periodic solutions.

The tethered system is modeled as a rigid dumbbell composed of two end masses and a straight tether spinning within an arbitrary spin plane. The attitude is parameterized by a 3-1-3 sequence of Euler angles that define the body frame with respect to an inertial frame. From this parameterization and assuming the tether is infinitely thin, a set of four first-order ordinary differential equations is found to describe the evolution of the three Euler angles and the spin rate. Assuming that the  spin rate is much larger than the orbital angular velocity, the tether spin angle can be averaged over a tether revolution. This operation removes the spin angle from the equations and decouples the evolution of the spin rate, yielding a non-autonomous set of ordinary equations for the two Euler angles that orient the spin plane with respect to the inertial frame. The averaged model is verified by comparing its solutions with those of the full system over a range of spin rates, showing that the mean relative error decreases rapidly as the spin rate increases. 

Analytical and numerical studies of the averaged equations show that the system possesses two equilibrium configurations, which correspond to the spin plane being equal to the orbital plane. Since the dynamics are studied in the inertial frame, the gravitational potential depends on time through the true anomaly. Nonetheless, the Jacobi invariant exists and it is used to demonstrate that the equilibria are Lyapunov-stable centers using Lyapunov's direct method. Finally, the dynamics of the spin plane as a function of the orbit inclination are analyzed by integrating the equations numerically and plotting the trajectories in the phase space. It is found that, for Sun-synchronous orbits, the optimal spin plane maximizing the semi-major axis variation lies close to a periodic solution of the averaged system, leading to very small oscillations of the spin plane around a nearly optimal orientation. These results suggest that, for moderate spin rates, the averaged model can reliably capture the slow evolution of the spin plane close to this particular solution. For an EDT---for which the effect of the Lorentz force should be considered---this periodic orbit may be particularly interesting to develop control strategies modulating the tether current.

REFERENCES
[1] PELAEZ, J.; LORENZINI, E. C.; LOPEZ-REBOLLAL, O.; RUIZ, M. A New Kind of Dynamic Instability
in Electrodynamic Tethers. Journal of the Astronautical Sciences. 2000, vol. 48, no. 4, pp. 449–476.
Available from doi: 10.1007/BF03546266.
[2] SIMON-AZNAR, J.; SANCHEZ-ARRIAGA, G.; VATANKHAHGHADIM, B. Optimal Spin Plane and
Performance of Spinning Electrodynamic Tethers for Orbital Maneuvering. Submitted to Journal of Guidance, Control,
and Dynamics. 2025
[3] PELAEZ, J.; COIRA, M. Dynamics of Fast-Rotating Tethered Satellites. Monografias de la Real Academia
de Ciencias Exactas, Fisicas, Quimicas y Naturales de Zaragoza. 2009, vol. 32, pp. 75–83. issn 1132-6360.