# Dynamic Behavior Analysis and Stability Control of Tethered Satellite Formation Deployment
###### Abstract  
In recent years, Tethered Space Systems (TSSs) have received significant attention in aerospace research as a result of their significant advantages: dexterousness, long life cycles and fuel-less engines. However, configurational conversion processes of tethered satellite formation systems in a complex space environment are essentially unstable. Due to their structural peculiarities and the special environment in outer space, TSS vibrations are easily produced. These types of vibrations are extremely harmful to spacecraft. Hence, the nonlinear dynamic behavior of systems based on a simplified rigid-rod tether model is analyzed in this paper. Two stability control laws for tether release rate and tether tension are proposed in order to control tether length variation. In addition, periodic stability of time-varying control systems after deployment is analyzed by using Floquet theory, and small parameter domains of systems in asymptotically stable states are obtained. Numerical simulations show that proposed tether tension controls can suppress in-plane and out-of-plane librations of rigid tethered satellites, while spacecraft and tether stability control goals can be achieved. Most importantly, this paper provides tether release rate and tether tension control laws for suppressing wide-ranging TSS vibrations that are valuable for improving TSS attitude control accuracy and performance, specifically for TSSs that are operating in low-eccentricity orbits. t +
Footnote †: journal: Sensors  
0000-0002-22226-1000/00000-0000-00000-00000-00000-0000-00000-00000-00000-00000-00000-000  
In another study, interorbital rendezvous with small relative inclination was also analyzed, and a nonlinear receding horizon controller was considered for tracking highly nonlinear systems by producing disturbances in system mass distributions and perturbations to initial system conditions [22]. [PERSON] and [PERSON] [23] proposed nonlinear feedback tension control laws based on [PERSON]'s method, used a linear state variable feedback control and affirmed the desired length of extended tethers in a reasonable amount of time. Stability analysis is a core research focus in mechanism studies of dynamic systems. The Floquet theory is a stability theory of solutions of linear ordinary differential equations with periodic variable coefficients [24] that was proposed by [PERSON] in 1868. Few researchers used the Floquet theory to study TSS stability and dynamic behavior. [PERSON] et al. [25] analyzed the spinning stability of a three-body Tethered Satellite Formation (TSF) by using Floquet theory, and stability analysis indicated that unstable motion occurs if its spinning angular rate is less than the critical value \(|-2.8|\) or \(0.65\) times its orbital angular rate. In another study, an analytical tether length rate control was designed, and parameter regions for stable deployment in order to maintain a tensile tether state were obtained [26]. [PERSON] and [PERSON] [27] analyzed the stability of out-of-plane vibrations of a spinning TSS, and two satellites as point masses were connected by a rigid rod, constraining the system's mass center to a circular orbit. The aforementioned references indicate that little attention has been given to numerical studies on the accuracy of simplified TSS models, control stability and the influence of orbital eccentricity. However, tethers produce in-plane and out-of-plane swings and longitudinal and transverse vibrations as a result of complex perturbations [28; 29], and a slight change in orbital eccentricity can significantly affect the original system. Previous studies show that analytical solutions for complex nonlinear models of TSS are difficult to obtain. This paper evaluates the stability of periodic TSS motions by using Floquet theory and provides control laws and small parameter domains of stability based on a simplified rigid-rod tether model. In this paper, nonlinear dynamic behavior and stability of TSS during deployment are analyzed.