Three-dimensional deployment of electro-dynamic tether via tension and current control with constraints
Introduction
The technology of space tether has drawn great attention from academia and industry over the past decades, due to its promising applications in space debris mitigation, orbit transfer, and deep space exploration, etc. [1], [2]. In a Space Tether System (STS), two or more end-bodies (e.g. satellites) are connected to each other via long tethers so as to perform cooperative tasks. Specifically, an STS with conductive tether is also known as an Electro-Dynamic Tether (EDT) system [1]. The electric current flowing in an EDT will interact with the geomagnetic field such that the orbit and attitude of the EDT system can be altered using the resulted Lorentz force without the consumption of chemical propellants. Recent research efforts of EDTs have mainly focused on their applications for deorbiting defunct satellites, where environmental perturbations have long-term effects on the orbital motion. For example, Zhong and Zhu investigated the long-term dynamics of deorbiting a nano-satellite by using a bare and short EDT with the consideration of the orbital environmental perturbations due to electrodynamic effect, atmospheric drag, and Earth's oblateness [3]. Li and Zhu proposed a nodal position finite element tether model for the dynamic simulation of spacecraft deorbit process using EDT over a long period, and applied a symplectic integrator for enforcing the energy and momentum conservation of the discretized finite element model [4].
Various technical aspects of the STS concept have been successfully demonstrated through missions flown in-orbit [5], [6]. A prerequisite of any STS mission is to deploy its tether to a commanded length. From a viewpoint of control, it is not easy to accomplish the deployment task in a simultaneously fast and stable manner since varying tether length may excite significant libration and vibration of the space tether [1], [7]. The deployment control of a space tether is not only nonlinear and under-actuated by nature, but also subject to the physical constraint that the tether tension should be kept positive since the tether cannot support any compression.
It has been demonstrated that tension control is very efficient for stabilizing any in-plane motion of an STS by appropriately regulating tether tension via a reel mechanism [8], [9], [10], [11], [12], [13]. For example, Rupp made the first attempt to devise a tension control law, with the feedback of tether length and length rate, for damping tether libration in the orbital plane [8]. Sun and Zhu developed a linear fractional-order control law of tether tension for the deployment control problem of an STS [9]. It is noted that the afore-mentioned studies did not explicitly account for the requirement of non-negative tension. With system constraints and nonlinear dynamics explicitly taken into account, significant efforts have been made to address the deployment control problem of an STS by using optimization-based techniques. For instance, Steindl and Troger considered the in-plane deployment of an STS under the inequality constraints of tether tension and length rate, and proposed an optimal control strategy to determine the tension profile for the deployment task [10]. Williams applied a variety of optimality criteria to solve, in an open-loop sense, the in-plane deployment and retrieval trajectories of an STS with state and control constraints [11]. Although effective, the optimal control strategies based on numerical optimization are highly demanding in computational load. As a computationally inexpensive alternative, Wen et al. recently developed an analytical feedback control law for the in-plane deployment of an STS, where the tension constraint was explicitly accounted for by a special saturation function [12].
The problem of deployment control is much more challenging in the presence of out-of-plane motion. As shown in Ref. [14] by using a dumbbell model, the out-of-plane libration of tether is almost decoupled from the other two degrees of freedom in the case of small state errors. Consequently, pure tension or length control is not enough for completely diminishing the three-dimensional libration of tether in a short time [2], [11], [15]. It has been proposed to augment tension control with out-of-plane thrust so as to gain better control performance for the three-dimensional deployment of STS. For example, Kumar and Pradeep designed linear feedback control strategies of tether tension and out-of-plane thrust for the three-dimensional deployment of an STS, and analyzed the controller stability by linearizing the dynamic equations of system around the system equilibrium point [16]. Wen et al. presented a second-order differential inclusion formulation for the nonlinear optimal control of an elastically tethered sub-satellite with out-of-plane thruster, and solved the open-loop optimal control inputs and state trajectories for the three-dimensional deployment process of the tether [17]. Notably, a propellantless means specifically possible for an EDT system is to apply Lorentz force for controlling the three-dimensional libration motions by regulating the electric current flowing in the EDT. For instance, Williams designed an energy rate feedback law of electric current for stabilizing the EDT librations around a prescribed periodic solution [18]. Larsen and Blanke modeled an EDT system using the port-controlled Hamiltonian formulation and proposed a passivity-based control law for the stabilization of the unstable motions in the EDT attitude [19].
This paper aims to achieving the three-dimensional deployment of an EDT system that consists of a main-satellite and a sub-satellite via the feedback control of tether tension and electric current, thereby without the consumption of chemical propellants. The deployment task of concern is to maneuver the tethered sub-satellite from an initial position close to the main-satellite to a desired radial equilibrium position far away from the main-satellite. To the best knowledge of authors, no attempt has been made to develop a three-dimensional deployment control law in an analytical form that can explicitly account for the control constraints of tether tension and electric current. To achieve this goal, the tension control law proposed in [12] for the in-plane deployment of an inert tether is extended via the augmentation of regulating the electric current flowing in the EDT. The proposed control strategy explicitly and analytically accounts for the control constraints by using a pair of special saturation functions, thereby involving much less computational loads than optimization-based strategies for constrained control problems.
The rest part of the paper is organized as follows. In Section 2, the dynamic equation of the EDT system is established. Then, the design and stability analysis of the controller are given in Section 3. In Section 4, the performance of the proposed control strategy is evaluated via numerical case studies. Finally, the conclusions are drawn in Section 5.
Section snippets
Dynamic model
In the EDT system of concern, a main-satellite is connected to a sub-satellite through a conductive tether, as shown in Fig. 1. The dynamics of the EDT system is modeled under the dumbbell assumption [1], where the tether flexibility and the satellite attitudes are neglected. Accordingly, the tether is treated as a massless straight link with a variable length , whereas the main-satellite and sub-satellite are simplified as two lumped masses and , respectively. The Center of Mass
Controller design and stability analysis
An analytical feedback control law for the three-dimensional deployment of the EDT system is synthesized in this section so as to stabilize the system motions by regulating the tension and electric current in the tether. The proposed controller can be decomposed into two parts, one is for tension control and the other is for current control. The first part is a basic tension control law that was originally proposed in Ref. [12] for controlling the in-plane deployment motion only. The tension
Case studies
The numerical simulation on a representative EDT system is given in this section to evaluate the performance of the proposed control strategy for the three-dimensional deployment task. Four case studies with different actuating conditions or orbital inclinations are presented for the sake of comparison. Tension control is solely applied to the first case with the electric current switched off, whereas the proposed control strategy is used in the other three cases for simultaneously regulating
Conclusions
This paper presents a propellant-free control strategy for the three-dimensional deployment of an EDT system. To gain better control performance of three-dimensional motions, the tension control law for the in-plane deployment of an inert tether is extended via the augmentation of regulating the electric current flowing in the EDT. In particular, the physical bounds of the tether tension and electric current are taken into consideration from the view point of practical implementation. To gain a
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grant 11002068 and 11372130, and in part by the Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant 201233.
References (25)
- et al.
Deployment and retrieval of tethered satellite system under J2 perturbation and heating effect
Acta Astronaut.
(2010) Dynamics and control of tethered satellite systems
Acta Astronaut.
(2008)- et al.
Strategies for three dimensional deployment of tethered satellites
Mech. Res. Commun.
(1998) - et al.
Advances in dynamics and control of tethered satellite systems
Acta Mech. Sin.
(2008) Review of dynamics and control of nonelectrodynamic tethered satellite systems
J. Spacecr. Rockets
(2006)- et al.
Dynamics of nanosatellite deorbit by bare electrodynamic tether in low earth orbit
J. Spacecr. Rockets
(2013) - et al.
Long-term dynamic modeling of tethered spacecraft using nodal position finite element method and symplectic integration
Celest. Mech. Dyn. Astron.
(2015) - M.L. Cosmo, E.C. Lorenzini, Tethers in Space Handbook, 3rd ed., NASA,...
- et al.
Optimal control of nanosatellite fast deorbit using electrodynamic tether
J. Guid., Control, Dyn.
(2014) - C.C. Rupp, A tether tension control law for tethered subsatellites deployed along local vertical, NASA-TM-X-64963,...
Fractional-order tension control law for deployment of space tether system
J. Guid., Control, Dyn.
Optimal control of deployment of a tethered subsatellite
Nonlinear Dyn.
Cited by (30)
Reinforcement learning-based attitude control for a barbell electric sail
2024, ISA TransactionsCross-verification and benchmarking analysis of electrodynamic tether simulators
2023, Acta AstronauticaLong-term deorbiting control for an electrodynamic tether system exploiting periodic solutions
2023, Acta AstronauticaCitation Excerpt :However, an EDT system with rigid or flexible tether on an inclined orbit is inherently unstable with a non-zero tether current, if the current is not explicitly controlled for orbital maneuvers like deorbiting [16]. Due to the continuous energy injection through the electrodynamic forces into the system, the EDT on inclined orbits may perform unstable librations with large amplitudes during deorbit [17]. The attitude dynamics and orbital dynamics of the EDT system are coupled through nonlinearities.
A code for the analysis of missions with electrodynamic tethers
2022, Acta AstronauticaCitation Excerpt :Secondly, accurate and flexible simulation tools are necessary to assess EDT’s performance and allow non-experts and potential investors to evaluate the technology, explore the use of EDTs in new scenarios, and make decisions. Accordingly, research groups and companies have developed EDT simulators with different degree of accuracy, complexity and computational cost like TEMPEST [16], FLEX [17], DYNATETHER [18], TetherSim™ [19], and BETsMA [20] among many others [21–28]. A central element of these software is the electric module, i.e. the numerical algorithm in charge of computing the current and voltage profiles throughout the tether.
Spin dynamics of a long tethered sub-satellite system in geostationary orbit
2022, Acta AstronauticaCitation Excerpt :Jung et al. [5] developed a two-piece dumbbell model with a moving mass and investigated the global tendencies of the libration angle difference with respect to the changes in the system parameters. Wen et al. [6] developed a dynamic model based on the classic dumbbell model and designed a three-dimensional deployment control law that can explicitly account for the control constraints of the tether tension and electric current. Li et al. [7] proposed a multiphysics finite element method, where the displacements of the tether is replaced by the nodal positions and studied the coupled effect between the motional electric field and the tether dynamics.