Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1986374, 7 pages

https://doi.org/10.1155/2017/1986374

## Attitude Motion of Cylindrical Space Debris during Its Removal by Ion Beam

Samara National Research University, 34 Moskovskoye Shosse, Samara 443086, Russia

Correspondence should be addressed to Vladimir S. Aslanov

Received 23 June 2017; Revised 4 October 2017; Accepted 8 November 2017; Published 26 November 2017

Academic Editor: Viktor Avrutin

Copyright © 2017 Vladimir S. Aslanov and Alexander S. Ledkov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper is devoted to the problem of space debris mitigation. Contactless method of the space debris deorbiting is considered. It is assumed that ion thrusters on the active spacecraft create the ion flow, which blows the debris and slows it down. The objectives of this work are the development of mathematical models and the research of space debris motion under the action of the ion flow. It is supposed that the space debris is a rigid body of a cylindrical shape. Calculation of ion beam force and torque was performed for a self-similar model of plasma plume expansion using the hypothesis of ion fully diffused reflection from a surface. A mathematical model describing plane motions of the cylindrical space debris under the influence of gravity gradient torque and the ion flux was constructed. It was shown that motion of the space debris around its center of mass has a significant effect on its removal time. Phase portraits, describing the motion of the space debris relative to its center of mass, were constructed. Comparison of the descent times in different motion modes was carried out. The results can be used to create new effective systems of large space debris removal.

#### 1. Introduction

During recent years in the scientific literature, considerable attention has been paid to the problem of transportation of nonfunctioning satellites and space debris removal. The majority of works were devoted to systems that imply a stage of docking or capturing a transported object by harpoons [1], a net [2], a tether [3], or a robotic manipulator [4]. A detailed overview of the capture tools and methods is given in the work [5]. Docking with an unmanaged object is a complex technical task. Failure at this stage is highly probable and can lead to the formation of new debris. An alternative is the use of noncontact transportation methods: based on the Coulomb interaction [6] and the ion beam created by the electric-reactive engine [7].

The use of the ion beam involves the placement of electric-reactive engines on an active spacecraft. These engines are not something exotic and are widely used in modern aerospace [8]. The engines “blow” on the transported object and thus change the parameters of its motion. To date, “Ion Beam Shepherd” is the best-designed project in this field. Existing studies show that the considered method allows removing an object with a mass of several tons from the orbit with a height of the order of 1000 km for several months [7]. Works [9, 10] are devoted to the estimation of the influence of the ion beam created by an electric-reactive engine, on objects of various forms. There are studies in which the optimal control laws of the active spacecraft for the space debris removal from orbit are developed [11, 12]. An interesting concept for detumbling spinning debris objects using the interaction between the thruster exhaust gases from the chaser and the debris object was considered in [13]. The authors have developed control function for detumble Envisat in a closed-loop simulation. Analysis of the literature shows that in existing studies due attention is not paid to the motion of the transported object relative to its center of mass.

The aim of this work is the development of mathematical models and research space debris motion under the action of the ion flow. It is supposed that the space debris is a rigid body of a cylindrical shape. The geometric parameters of the cylinder, which was taken as an example in this paper, correspond to the Cosmos 3M rocket stage. Currently, about 300 such stages are in orbit in some of the most crowded orbital regions [14]. The removal of these objects from the orbits is of great importance for the space debris mitigation issue.

#### 2. Materials and Methods

Mathematical models of a space debris plane motion under the action of the gravitational, ion flux forces, and torques will be developed in this section. Method of the ion beam resultant force and torque calculation will also be described.

##### 2.1. Equations of Plane Motion

The plane motion of space debris is considered. It is assumed that the space debris is a rigid body; the Earth does not rotate, and it has a spherical shape; the active spacecraft, which creates ion flow, is maintained in a fixed position relative to the center of mass of the space debris by its control system. It is supposed that only gravitational and ion forces and torques act on space debris.

Let us introduce the inertial coordinate system . Origin* O* is the center of the Earth. The axis passes through the pericenter of initial space debris orbit. The origin of the orbital frame is located at the center of mass of the space debris (Figure 1). The axis lies along the radius vector of the space debris center of mass. The axis is directed towards the orbital flight. The body frame is fixed relative to the space debris. The active spacecraft is a material point (Figure 1). Its radius vector has the following coordinates: in frame. The direction of the ion beam axis is defined by the angle . The axis is directed along ion beam axis to flight direction, and the axis completes the right-handed set.