Tethered Satellite Formation Research Articles

Full State Estimation for Triangular Tethered Satellite Formations: Filter Design, Observability Analysis, and Performance Evaluation

Full State Estimation for Triangular Tethered Satellite Formations: Filter Design, Observability Analysis, and Performance Evaluation

Chain tethered satellite formation reconfiguration: An event-triggered practical prescribed-time control approach

This paper addresses the challenge of rapid reconfiguration control in the chained tethered satellite formation (CTSF) amidst full-state constraints and uncertainties. To tackle this, we propose an innovative event-triggered practical prescribed-time control method grounded in observer theory. Initially, we develop a sliding mode state observer utilizing event triggering to effectively monitor the velocities of adjacent satellites. Subsequently, leveraging the framework of backstepping control, uncertainties are managed by approximating them using fuzzy logic systems. The imposition of full-state constraints is adeptly handled through the barrier Lyapunov function. An event-triggered practical prescribed-time controller (ET-PPTC) is then engineered, drawing on the principles of practical prescribed-time stability. The robustness and stability of the closed-loop control system are rigorously established employing the Lyapunov framework. Finally, through comprehensive numerical simulations, we validate the feasibility and efficacy of the proposed control scheme in facilitating the reconfiguration mission of a four-satellite CTSF.

Deployment Dynamics and Control of a Hub-Spoke Tethered Satellite Formation Using Combined Arbitrary Lagrange-Euler and Referenced Nodal Coordinate Formulation

Abstract A novel modeling framework combining arbitrary Lagrange-Euler and referenced nodal coordinate formulation (ALE-RNCF) is proposed for deployment dynamics and control of a hub-spoke tethered satellite formation. The ALE-RNCF approach allows for an accurate analysis of the intricate coupling effect between the orbit, attitude, and deployment dynamics, and its strengths lie in overcoming the accuracy loss and low-efficiency issues when dealing with spatial and temporal multiscale problems. Specifically, the orbital and attitude motions are separated with vibrations of the variable-length ALE tethers through the RNCF, which is the main distinguishing feature over the widely-used absolute nodal coordinate formulation. To achieve stable deployment, the control torque is added to the central satellite by employing the proportional-differential algorithm, where the maximum tension of tethers or the spinning angular velocity is selected as the control object. Various cases with different deployment velocities, target tensions, and orbital heights are simulated and corresponding effects on the deployment performance are analyzed. The proposed ALE-RNCF approach provides a comprehensive understanding of the orbit-attitude-structure coupled behavior during the deployment of the hub-spoke tethered satellite formation and contributes to the development of effective control strategies.

Stability Analysis and RBF Neural Network Control of Second-Order Nonlinear Satellite System

This paper studies the open-loop stability analysis and radial basis function based neural-network(RBFNN) control of the triangular tethered satellite formation(TTSF) systems which comprise the in-plane and out-plane dynamics. The dynamical equations of TTSF are nonlinear coupled second-order differential equations(DE), whose explicit solutions are hard to work out. Therefore, the dynamics of systems are analyzed by imposing some constraints on initial states, and the norms of proper initialization for such TTSF systems are obtained by solving the DE. As a result, the nonlinear controllers achieving stable tracking to reference states of TTSF systems are designed based on open-loop analysis. Furthermore, when the structure information of the system is uncertain, delay sampled NN(DSNN) as well as offline training NN(OTNN) control algorithms are proposed to train RBFNN, and the optimal weights of NN are computed by least square(LS) method. Then RBFNN control is imposed to the uncertain TTSF system to ensure the stable tracking, and the stability proof of the control is given. Finally, simulations are taken on the specific motions of TTSF system to test the theoretical results.

Minimalistic estimation with stochastic constraints for closed-type tethered satellite formations

This paper aims to propose a constrained estimation scheme with minimalistic sensor settings for a class of closed-type tethered satellite formation (TSF). The estimation scheme is built on the fusion of dynamic model, priori tether length constraints, and position observation through a learning-based constrained particle filtering (CPF) design. The CPF characterizes the tether lengths as stochastic constraints and integrates these constraints within Bayesian framework based on pseudo observation and radial basis function (RBF) neural network. The RBF neural network can learn the underlying relationship between the system states and uncertain tether oscillations to establish a probability distribution function (PDF) of stochastic constraints. Nonlinear observability analysis using observability rank criterion is performed to yield insight into the minimum number of positioning sensors needed for n-body (n≥3) closed-type TSF. Furthermore, it is found that the use of pseudo observation for filtering can reduce the number of needed positioning sensors. Interestingly, the minimum number of positioning sensors required for determining all position information of n-body (n≥3) closed-type TSF is 1. Finally, the effectiveness of proposed estimation scheme is verified by posterior Cramér-Rao lower bound (PCRLB) calculation and extensive simulations. The proposed estimation scheme can be readily extended to other types of TSF. For example, the chain-type TSF.

Observer‐based adaptive backstepping control and the application for a class of multi‐input multi‐output nonlinear systems with structural uncertainties and perturbations

AbstractIn this study, the adaptive backstepping control of a class of multi‐input multi‐output nonlinear systems with immeasurable states, structural uncertainties and periodic perturbations is researched using neural network (NN) based nonlinear observer. Firstly, a series of harmonic components obtained from Fourier series expansion (FSE) are employed to estimate the unknown periodic perturbations. Then treating the estimated perturbations as inputs, a radial basis function‐based neural network (RBFNN) is constructed as a component of the observer, and the observer is proved to be uniformly ultimately bounded (UUB) in estimating the immeasurable system states with the negative‐gradient adaptive laws of NN parameters. Subsequently, the adaptive backstepping control is designed to track reference signals at the outputs of system based on the FSE‐RBFNN observer, and the stability of the closed‐loop system is proved on a finite set of system states. Finally, the proposed methods are applied to a triangular tethered satellite formation model to test the stability of the observer and control, and the effect of FSE in estimating perturbations is comparatively tested as well.

Stability Research of a Triangular Tethered Satellite Formation: Dynamics, Filtering, and Control

This article investigates the stability of a triangular tethered satellite formation (TTSF) and the performance improvement by filtering and the virtual control of general force. The equation of motion is formulated by Lagrangian equation, and the state space equation (SSE) is derived with the second-order terms decoupled. Then the stability of SSE is discussed with the tethers nonstretched and stretched, respectively, where the existence of a stable orbit arounthe equilibrium is explored and the effect of initial conditions on the stability of the system around a stable orbit is discussed. To smooth the system response, a steepest descent-based state filter for TTSF system is designed, which provides the optimized state information. Moreover, to get the expected system response, a virtual control law is acted on the system with hysteresis relay property employed to strengthen its robustness. Finally, simulation results are shown to validate the theoretical analyses.

Spin deployment of Hub-Spoke tethered satellite formation with sliding mode tether tension control

This work develops a tension control strategy for deploying an underactuated spin-stable tethered satellite formation in the hub-spoke configuration. First, the Lagrange equation is used to model the spin-deployment dynamics of the tethered satellite formation. The central spacecraft is modeled as a rigid body, and the tethered subsatellites are simplified as lumped masses. Second, a pure tension controller has been proposed to suppress the tether libration motion in the deployment without thrusting at the subsatellites. A nonlinear sliding mode control is introduced in the tension controller for the underactuated system to suppress the periodic gravitational perturbations caused by the spinning hub-spoke tethered satellite formation. The unknown upper bounds of the perturbations are estimated by adaptive control law. The bounded stability of the closed-loop tension controller has been proved by the Lyapunov theory. Finally, numerical simulations validate the effectiveness and robustness of the proposed controller, i.e., tethers are fully deployed stably to the desired hub-spoke configuration.

Nonlinear State Observer and Control Design for Triangular Tethered Satellite Formation

This article researches the control of triangular tethered satellite formation (TTSF) system and nonlinear state observer (NSO) design when velocity vector is unmeasured. First, the stability around the equilibrium of open-loop system is discussed, which is stable around certain equilibrium pairs. To improve the system performance, linear feedback control is designed using Jacobian matrix of original nonlinear system, which is locally stable around linearized point. Besides, nonlinear control is designed subsequently which owns global stability domain. Then in case of the unmeasured velocity states, NSO is designed which contains a nonlinear function of estimation error to regulate the observation bias. With the designed nonlinear control and NSO, the simulation of the deployment of TTSF is made thereafter. Compared with linear feedback control and general observer with linear regulation, the proposed nonlinear control together with NSO has global stability domain and less chattering, which is promising in practical application.

Dynamic Behavior Analysis and Stability Control of Tethered Satellite Formation Deployment.

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.

Robust control of triangular tethered satellite formation with unmeasured velocities

This article researches the stable deployment of a triangular tethered satellite formation (TTSF) with velocities unmeasurable. By state space transformation, the system is transformed to four subsystems. Controllers with two kinds of robust terms are proposed and proved to make the system asymptotically stable. Then a high gain observer is given to estimate unmeasurable velocity variables in the system, where the design principles of the observer gains are found as well, and the system is locally asymptotically stable for the nonlinearities in the observer. Furthermore, the factors affecting the convergence of the system in the presence of state observer are found by virtue of Lyapunov function. Numerical simulation validates the effectiveness of the proposed controller and the stability of the system with state observer.

State estimation of double-pyramid tethered satellite formations using only two GPS sensors

Recently, space tethered satellite system has attracted widespread attention due to its attractive potential in space applications. Significant research on the tethered satellite formations has been undertaken to study the dynamics and control, vibration modes, tether models, and tether applications and it is generally assumed that the state information of the tethered satellite system is completely known or can be easily measured. However, few studies have been presented on the state estimates of the tethered formations. In this study, a state estimation scheme, with a minimal sensor fusion design, is presented for double-pyramid tethered satellite formations. The objective is achieved by fusing only the measurements of the two GPS sensors and the tether constraints. Furthermore, the observability of the proposed fusion design is proved through the analyses of Lie derivatives. Simulation results are presented that validate the proposed scheme. The proposed scheme provides a general hint of state estimation for a class of satellite formations with the tethered constraints.

Analysis of the deployment of a three-mass tethered satellite formation

Tethered satellite formation has obtained widespread attention in recent years. The main causes of the increasing study interest around tethered formation lie in its promised applications in space, such as interferometric measurements. The deployment of a tethered formation system in low Earth orbit is investigated in this paper. The orbital tethered system consists of three nanosatellites connected via tethers end-to-end, and the desired arrangement for the end masses is an equilateral triangle. For the sake of brevity, the formation system is modeled as a particle-rigid rod system, in which the elasticity of the tethers is omitted. The deployment process is carried out using tension forces and active external forces generated by low-thrust engines. The numerical results confirm that it is possible to deploy a triangular tethered system, which rotates at a given initial angular velocity, to an expected arrangement using the proposed control law of tether tensions and active forces.

A state estimation scheme with minimal sensor configuration for three-body tethered satellite formations

Tethered satellite formations in space have attracted widespread attention in recent years, as they can provide an excellent large-scale, self-stabilized space platform. At present, most studies on tethered satellite formation systems focus on dynamics, control deployment and stability analysis, and they all assume that the state information of the system is given or easily measured. However, it is not trivial to obtain accurate state measurements of tethered satellite formations in space, especially without using high-cost sensors. In this paper, we present a filter-based state estimation scheme with minimum sensor configuration for three-body tethered satellite formation. The satellite-to-satellite ranges and absolute attitude angles of tethered formation can be accurately estimated by fusing only the measurements of three gyroscopes and three tether releasing mechanisms. Particularly, the absolute attitude angles are obtained even without using any sensors that can measure absolute information. The feasibility of the proposed scheme is theoretically proven via the observability analysis of nonlinear systems, and the performance of the scheme is demonstrated through extensive simulations. This method can be readily extended to the state estimation process of other tethered formations in space.

Dynamics of a tethered satellite formation for space exploration modeled via ANCF

This paper studies the dynamics of a spinning three-body tethered satellite formation by a flexible multibody dynamics method. The tethered system is specially designed to constitute a Space Infrared Interferometric Telescope for a space exploration mission. Two sub-satellites are deployed from the main-satellite by using long flexible tethers and moving along trajectories of Archimedes spirals. Both of the large displacement and large deformation of the variable-length tethers are described by utilizing the Absolute Nodal Coordinate Formulation method in a framework of Arbitrary Lagrange-Euler (ANCF-ALE) description. The Natural Coordinate Formulation (NCF) method is used for describing the overall rigid motions of the three satellites. Considering the mass flow of tether element, the dynamic equations of tether element are derived by using the D'Alembert's principle. The governing dynamic equations of the system are obtained with a method of Lagrange multipliers. The tether deployment is achieved by regulating deployed velocity and applied jet forces. Numerical case studies are presented for investigating the parametric influences on the deployment dynamics of the tethered system by comparisons.

Stability and ground experiments of a spinning triangular tethered satellite formation on a low earth orbit

This paper studies the spinning stability of a triangular tethered satellite formation that flies on a low earth orbit. The spinning around the center of mass at a constant angular rate gives rise to a periodic motion in the non-inertial orbital frame. Floquet theory is used to analyze the stability of the periodic motion. A dynamic similarity between the on-orbit dynamics and ground experimental models is built to construct an equivalent ground experiment to verify the stability analysis. The analytical and experimental results show that a stable periodic motion can be guaranteed if the spinning angular rate of the system exceeds a critical value.

Sliding mode control of electromagnetic tethered satellite formation

This paper investigates the control of tethered satellite formation actuated by electromagnetic dipoles and reaction wheels using the robust sliding mode control technique. Generating electromagnetic forces and moments by electric current coils provides an attractive control actuation alternative for tethered satellite system due to the advantages of no propellant consumption and no obligatory rotational motion. Based on a dumbbell model of tethered satellite in which the flexibility and mass of the tether is neglected, the equations of motion in Cartesian coordinate are derived. In this model, the J2 perturbation is taken into account. The far-field and mid-field models of electromagnetic forces and moments of two satellites on each other and the effect of the Earth’s magnetic field are presented. A robust sliding mode controller is designed for precise trajectory tracking purposes and to deal with the electromagnetic force and moment uncertainties and external disturbances due to the Earth’s gravitational and magnetic fields inaccuracy. Numerical simulation results are presented to validate the effectiveness of the developed controller and its superiority over the linear controller.

Tethered satellite system control using electromagnetic forces and reaction wheels

In this paper a novel non-rotating space tethered configuration is introduced which its relative positions controlled using electromagnetic forces. The attitude dynamics is controlled by three reaction wheels in the body axes. The nonlinear coupled orbital dynamics of a dumbbell tethered satellite formation flight are derived through a constrained Lagrangian approach. These equations are presented in the leader satellite orbital frame. The tether is assumed to be mass-less and straight, and the J2 perturbation is included to the analysis. The forces and the moments of the electromagnetic coils are modeled based on the far-filed model of the magnetic dipoles. A guidance scheme for generating the desired positions as a function of time in Cartesian form is presented. The satellite tethered formation with variable length is controlled utilizing a linear controller. This approach is applied to a specified scenario and it is shown that the nonlinear guidance method and the linear controller can control the nonlinear system of the tethered formation and the results are compared with optimal control approach.

Nonlinear dynamics of a rotating triangular tethered satellite formation near libration points

The nonlinear coupled dynamics of a rotating triangular tethered satellite formation near libration points is presented. Based on Hill's approximation, the dynamical formulation for a rotating triangular tethered system near libration points is developed. The formulation comprises a set of strong nonlinear coupled equations. Numerical simulations using the full nonlinear and time-varying motion equations are performed for the station-keeping stages. The dynamic characteristics of the triangular tethered formation are analyzed. The numerical results demonstrate that the rotation rate has a significant impact on the stability of the system, whereas the orbital amplitude has a moderate impact on the stability of the system. Tether length has little effect on the motion of the system's center of mass. It can be served as a basis to solve the problem of rotating triangular tethered systems in any arbitrary plane.

Deployment and retrieval of a rotating triangular tethered satellite formation near libration points

The dynamic stability of a rotating triangular tethered satellite formation near libration points during the deployment and retrieval stages is investigated. Based on Hill׳s approximation, a new dynamical formulation for the attitude and orbital motions of the system is developed. Using numerical simulations, parametric studies of the effects of the orbital amplitude, initial rotation rate, tether length, and length rate are performed. It is shown that different initial rotation rates or length rates have almost no impact on the trajectories of the system׳s centroid. As expected, increasing the initial rotation rate or reducing the length rate can improve the stability of a tethered satellite system.