Passive Vertical Stabilization of Two Tethered Nanosatellites with Engineered Damping
AIAA/AAS Astrodynamics Specialist Conference and Exhibit
A simple method for the vertical stabilization of two tethered nanosatellites in low Earth orbit is described. Without active attitude control or passive damping, the satellite system would freely oscillate like a pendulum due to the gravity gradient moment acting differentially on the displaced satellites. It would also vibrate axially like a bouncing ball as the tether cycles from tautness to slackness. What ambient friction exists within the satellite system might eventually damp out these
... ly damp out these librations and vibrations, but with small space structures such dissipative mechanisms are meager: undesirable oscillations could persist for months. The performance benefit of engineering a passive spring-damper mechanism along the length of the tether connecting the two nanosatellites is investigated. Two-dimensional orbital-plane analysis only is given in this paper. The axial spring-damper is tuned by frequency matching the axial and in-plane librational modes so that they are coupled as strongly as possible. The coupling encourages the two modes to exchange energies. Over time, net energy is removed from the librations and dissipated by the axial spring-damper mechanism. The system is driven towards stable equilibrium. It is shown that a stiff spring is inadequate; there must be significant play in the tether when it is in tension for the scheme to work. The tuned parameters are shown to perform well over a broad range of initial conditions, from deployment to near stable equilibrium. Except when initialized near the unstable equilibrium, the method reduces the amplitude of in-plane librations by a factor of five in about one day for most initial conditions. The axial vibrations are seen to attenuate in unison with the librations. The 1 km tether, however, failed to fully deploy. At Stanford University's Space Systems Development Laboratory (SSDL), our primary mission is to enable nanosatellites to achieve more relevant scientific missions with minimal increase in complexity. One area where nanosatellites have room for improvement is attitude control. Recent missions have relied on passive magnetic stabilization to maintain alignment with the Earth's magnetic field, but this forces the satellite to tumble at twice the orbital rate. For many missions, a nadir-pointing satellite is a requirement, either for a scientific payload to study the Earth or for a directional antenna to achieve the necessary bandwidth for communication. A tethered pair of nanosatellites could be deployed so that a vertical orientation is maintained with respect to the Earth by means of gravity gradient stabilization. 4 Tethered systems librate freely in orbit, as would any object possessing a nonspherical moment of inertia. 5 Long transients may persist in reaching vertical stabilization unless some means of damping is provided. Damping is easily mechanized along the length of the tether, in the axial direction between the two nanosatellites. But axial damping does not normally damp out librations; some means are required to damp out librations in a reasonable amount of time. Gravity gradient stabilization has successfully been used on much larger and more massive conventional satellites in Earth orbit. In larger satellite systems, internal effects such as fuel sloshing, flexing of booms, or thermal expansion and contraction may be sources of passive damping. But there is not much opportunity for such ambient damping to exist in a nanosatellite, thus there is more of a need to engineer the damping into a tethered nanosatellite system. Conventional satellites are typically rigid bodies, whereas tethered nanosatellites are not. There are, however, several similarities. Because of the natural centripetal force constantly forcing the two apart, the tethered satellites when near librational and axial equilibrium maintain tautness in the tether, thus acting much like a rigid body. Also when near the stable equilbrium, the equations of motion depend little on mass or tether length: the dynamical motion, natural frequencies and energy exchanges are nearly the same. Despite these similarities, nanosatellites have much smaller mass than conventional satellites, so are more susceptable to external disturbances. Early in the investigation, two possible damping mechanisms were identified: • Axial mechanism: the tether itself acts as a spring-damper in the axial direction; and • Rotational mechanism: a rotational spring-damper is implemented at the anchor points where the tether connects to the nanosatellites.