Advances in dynamics and control of tethered satellite systems

The concept of tethered satellite system （TSS） promises to revolutionize many aspects of space exploration and exploitation. It provides not only numerous possible and valuable applications, but also challenging and interesting problems related to their dynamics, control, and physical implementation. Over the past decades, this exciting topic has attracted significant attention from many researchers and gained a vast number of analytical, numerical and experimental achievements with a focus on the two essential aspects of both dynamics and control. This review article presents the historic background and recent hot topics for the space tethers, and introduces the dynamics and control of TSSs in a progressive manner, from basic operating principles to the state-of-the-art achievements.

Nonlinear dynamics of flexible tethered satellite system subject to space environment

The paper studies the nonlinear dynamics of a flexible tethered satellite system subject to space environments, such as the J2 perturbation, the air drag force, the solar pressure, the heating effect, and the orbital eccentricity. The flexible tether is modeled as a series of lumped masses and viscoelastic dampers so that a finite multi- degree-of-freedom nonlinear system is obtained. The stability of equilibrium positions of the nonlinear system is then analyzed via a simplified two-degree-freedom model in an orbital reference frame. In-plane motions of the tethered satellite system are studied numerically, taking the space environments into account. A large number of numerical simulations show that the flexible tethered satellite system displays nonlinear dynamic characteristics, such as bifurcations, quasi-periodic oscillations, and chaotic motions.

Dynamics and rebound behavior analysis of flexible tethered satellite system in deployment and station-keeping phases

The tether deployment of a tethered satellite system involves the consideration of complex dynamic properties of the tether,such as large deformation,slack,and even rebound,and therefore,the dynamic modelling of the tether is necessary for performing a dynamic analysis of the system.For a variablelength tether element,the absolute nodal coordinate formulation(ANCF)in the framework of the arbitrary Lagrange-Euler(ALE)description was used to develop a precise dynamic model of a tethered satellite.The model considered the gravitational gradient force and Coriolis force in the orbital coordinate frame,and it was validated through numerical simulation.In the presence of dynamic constraints,a deployment velocity of the tether was obtained by an optimal procedure.In the simulation,rebound behavior of the tethered satellite system was observed when the ANCF-ALE model was employed.Notably,the rebound behavior cannot be predicted by the traditional dumbbell model.Furthermore,an improved optimal deployment velocity was developed.Simulation results indicated that the rebound phenomenon was eliminated,and smooth deployment as well as a stable state of the station-keeping process were achieved.Additionally,the swing amplitude in the station-keeping phase decreased when a deployment strategy based on the improved optimal deployment velocity was used.

Review of deployment technology for tethered satellite systems

Tethered satellite systems（TSSs） have attracted significant attention due to their potential and valuable applications for scientific research. With the development of various launched on-orbit missions, the deployment of tethers is considered a crucial technology for operation of a TSS. Both past orbiting experiments and numerical results have shown that oscillations of the deployed tether due to the Coriolis force and environmental perturbations are inevitable and that the impact between the space tether and end-body at the end of the deployment process leads to complicated nonlinear phenomena. Hence, a set of suitable control methods plays a fundamental role in tether deployment. This review article summarizes previous work on aspects of the dynamics, control, and ground-based experiments of tether deployment. The relevant basic principles, analytical expressions, simulation cases, and experimental results are presented as well.

Adaptive neural dynamic-based hybrid control strategy for stable retrieval of tethered satellite systems

This study proposes a novel adaptive neural dynamic-based hybrid control strategy for stable subsatellite retrieval of two-body tethered satellite systems.The retrieval speed is given analytically,ensuring a libration-free steady state.To mitigate the potential libration motion,a general control input signal is generated by an adaptive neural-dynamic(AND)algorithm and executed by adjusting the retrieval speed and thruster on the subsatellite.To address the limited retrieval speed and improve the control performance,the thruster controller is manipulated according to a novel advanced state fuzzy control law based on higher-order libration states,whereas the remaining control input is allocated to the speed controller.The Lyapunov stability of the control strategy is demonstrated analytically.Numerical simulations validate the proposed control strategy,demonstrating well-allocated control inputs for both controllers and good control performance.

On the optimality of uniform velocity-deceleration separation scheme for tethered satellite systems

For mitigating the libration angle fluctuation of the tethered satellite system,this paper discusses how to make the uniform velocity-deceleration separation scheme achieve the best effect.First,a judgment condition is established to determine the tether state by comparing the tether length and the relative distance of the sub-satellite and the parent satellite.Based on the tethered satellite system dynamics equation and Clohessy-Wiltshire equation,dynamic models are given for four cases of tether states.Second,the influence of the uniform velocity-deceleration separation scheme on the libration angle is analyzed by taking the libration angle at the separation ending time and the mean absolute value of the libration angle as index functions.Then,the optimality problem of the uniform velocity-deceleration separation scheme is formulated as an optimization problem with constraints,and an approximate solution algorithm is given by combining the back propagation neural network and Newton-Raphson method of multiple initial values.Finally,the effectiveness of the proposed method is verified by a numerical simulation.

Nonlinear dynamics and station-keeping control of a rotating tethered satellite system in halo orbits

The dynamics of a rotating tethered satellite system （TSS） in the vicinity of libration points are highly nonlinear and inherently unstable. In order to fulfill the station-keep control of the rotating TSS along halo orbits, a nonlinear output tracking control scheme based on the θ- D technique is proposed. Compared with the popular time-variant linear quadratic regulator （LQR） controller, this approach overcomes some limitations such as on-line computations of the algebraic Riccati equation. Besides, the obtained nonlinear suboptimal controller is in a closed form and easy to implement. Numerical simulations show that the TTS trajectories track the periodic reference orbit with low energy consumption in the presence of both tether and initial injection errors. The axis of rotation can keep pointing to an inertial specific object to fulfill an observation mission. In addition, the thrusts required by the controller are in an acceptable range and can be implemented through some low-thrust propulsion devices.

Coupling dynamic characteristics of simplified model for tethered satellite system

connecting wires are the main manifestations of the coupling dynamic effects on the orbit evolution,the attitude adjusting and the flexible vibration of the tethered satellite system.To investigate attitude evolution of the tethered system and the mechanical energy transfer/loss characteristics between the bus system and the solar sail via the connecting wires,a structure-preserving method is developed in this paper.Simplifying the tethered satellite system as a composite structure consisting of a particle and a flexible thin panel connected by four special springs,the dynamic model is deduced via the Hamiltonian variational principle firstly.Then,a structure-preserving approach that connects the symplectic Runge-Kutta method and the multi-symplectic method is developed.The excellent structure-preserving property of the numerical scheme constructed is presented to illustrate the credibility of the numerical results obtained by the constructed structure-preserving approach.From the numerical results on the mechanical energy transfer/loss in the composite structure,it can be found that the mechanical energy transfer tendency in the tethered system is dependent of the initial attitude angle of the system while the total mechanical energy loss of the system is almost independent of the initial attitude angle.In addition,the special stiffness range of the spring is found in the attitude angle evolution of the system,which provides a structural parameter design window for the connecting wires,that is,the duration needed to arrive the stable attitude is short when the stiffness of the wire is designed in this special range.

Equilibrium configurations of the tethered three-body formation system and their nonlinear dynamics

This paper considers nonlinear dynamics of teth- ered three-body formation system with their centre of mass staying on a circular orbit around the Earth, and applies the theory of space manifold dynamics to deal with the nonlinear dynamical behaviors of the equilibrium configurations of the system. Compared with the classical circular restricted three body system, sixteen equilibrium configurations are obtained globally from the geometry of pseudo-potential energy sur- face, four of which were omitted in the previous research. The periodic Lyapunov orbits and their invariant manifolds near the hyperbolic equilibria are presented, and an iteration procedure for identifying Lyapunov orbit is proposed based on the differential correction algorithm. The non-transversal intersections between invariant manifolds are addressed to generate homoclinic and heteroclinic trajectories between the Lyapunov orbits. （3,3）- and （2,1）-heteroclinic trajecto- ries from the neighborhood of one collinear equilibrium to that of another one, and （3,6）- and （2,1）-homoclinic trajecto- ries from and to the neighborhood of the same equilibrium, are obtained based on the Poincar6 mapping technique.