Three-dimensional deployment of electro-dynamic tether via tension and current control with constraints

Highlights

• The proposed control strategy is given in an analytical form.

• The tension and current in the tether are simultaneously regulated.

• The tension and current constraints are explicitly accounted for.

Abstract

The concept of space tether has found a great deal of promising applications in space engineering. A prerequisite of any space tether mission is to deploy its tether to a commanded length. This paper aims to achieving the three-dimensional deployment of an electro-dynamic tether system in a propellant-free manner via the feedback control of the tension and electric current in the tether. The proposed controller is formulated in an analytical form with an extremely low level of computational load, and can explicitly account for the physical bounds of the tether tension and electric current by using a pair of strictly increasing saturation functions. In addition, the Lyapunov analysis is made to gain an insight into the stability characteristics of the proposed control strategy. To facilitate the theoretical analysis, the dynamic model of the system is developed under the widely used dumbbell assumption, along with the geomagnetic field modeled using a tilted dipole approximation. Finally, numerical case studies on a representative electro-dynamic tether system are conducted to evaluate the performance of the proposed controller and the influence of the actuating conditions and orbital inclinations.

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.