International Journal of Aerospace Engineering

Volume 2017, Article ID 9734164, 10 pages

https://doi.org/10.1155/2017/9734164

## Autonomous Orbit Determination for Lagrangian Navigation Satellite Based on Neural Network Based State Observer

^{1}College of Astronautics, Nanjing University of Aeronautics and Astronautics, Nanjing, China^{2}College of Astronomy and Space Science, Nanjing University, Nanjing, China

Correspondence should be addressed to Youtao Gao; nc.ude.aaun@oagty

Received 1 March 2017; Revised 11 May 2017; Accepted 24 May 2017; Published 21 June 2017

Academic Editor: Paolo Tortora

Copyright © 2017 Youtao Gao et al. 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

In order to improve the accuracy of the dynamical model used in the orbit determination of the Lagrangian navigation satellites, the nonlinear perturbations acting on Lagrangian navigation satellites are estimated by a neural network. A neural network based state observer is applied to autonomously determine the orbits of Lagrangian navigation satellites using only satellite-to-satellite range. This autonomous orbit determination method does not require linearizing the dynamical mode. There is no need to calculate the transition matrix. It is proved that three satellite-to-satellite ranges are needed using this method; therefore, the navigation constellation should include four Lagrangian navigation satellites at least. Four satellites orbiting on the collinear libration orbits are chosen to construct a constellation which is used to demonstrate the utility of this method. Simulation results illustrate that the stable error of autonomous orbit determination is about 10 m. The perturbation can be estimated by the neural network.

#### 1. Introduction

The navigation of deep space probes is one of the main problems that restrict the deep space exploration. Generally, navigation support for deep space probes is primarily provided by the NASA’s Deep Space Network (DSN). In order to improve the navigation performance by efficiently determining angular position of interplanetary spacecraft, the delta differential one-way tanging technique is also employed by some missions. Besides that, some autonomous navigation strategies are also proposed to support future deep space exploration missions. Autonomous navigation is important for deep space probes to deal with communication delay as well as reducing the dependency on ground stations. Several autonomous navigation methods have been proposed for deep space probes. As early as 1968, the sextant had been used for autonomous navigation in Apollo program [1]. In 1999, Deep Space 1 achieved autonomous orbit determination by tracking small celestial bodies with an optical sensor [2]. The comet probe “Deep Impact” carried out its navigation automatically based on an optical navigation system [3]. The rotation period of X-ray pulsar is extremely stable; therefore, time and the location of spacecraft have been proposed to be determined by tracking several X-ray pulsars in [4, 5]. The Global Positioning System (GPS) can support the navigation of deep space probes when they are orbiting in low Earth orbits and medium Earth orbits. For deep space transfer orbits and deep space target orbits, due to limited visibility, extremely low signal-to-noise ratio, and poor relative geometry among sources and users, GPS is not adequate. In 2005, Hill suggested placing navigation constellation on the periodic orbits in the vicinity of libration points of the Earth-Moon system to support deep space navigation [6]. Similar to GPS, a high-precision satellite navigation constellation which consists of libration point satellites in the Earth-Moon system is introduced to provide navigation information for deep space probes, which can be called, accordingly, the Lagrangian point satellite navigation system. The satellites which construct the Lagrangian point satellite navigation system are called Lagrangian navigation satellites. Zhang and Xu analyzed the architecture and navigation performance of the Lagrangian point satellite navigation system [7–9]. The Lagrangian navigation constellation is introduced to navigate the deep space probes autonomously. Hence the navigation constellation itself should have the ability of autonomous orbit determination (AOD). In [10], the feasibility of AOD for satellites in quasiperiodic orbits about the Earth-Moon libration point was verified. Based on circular restricted three-body problem (CRTBP), Du et al. researched the autonomous orbit determination method of satellites in halo orbits, and only satellite-to-satellite range was used as observation [11]. For the Earth navigation satellite constellation, there is a rank deficiency problem when only satellite-to-satellite range is used to determine the orbit. However, the rank deficiency problem does not exist for the Lagrangian navigation satellites because of the special dynamics near the libration points [6]. Thus, the Lagrangian navigation satellites in the navigation constellation can autonomously determine their orbits using only satellite-to-satellite range. In [12], Gao et al. discussed the algorithm of autonomous orbit determination using only the satellite-to-satellite range measurement for Lagrangian navigation constellation. The current studies about AOD of Lagrangian navigation satellites are under the CRTBP model. However, the motion of the Moon around the Earth is eccentric. Therefore, the elliptic restricted three-body problem (ERTBP) is more accurate to describe the motion of the Lagrangian navigation satellite [13]. The ERTBP has been discussed in detail in [14–16]. Our purpose here is to extend the applicability of ERTBP to the study of the AOD of Lagrangian navigation satellites. In this study, we consider ERTBP with perturbation. The perturbation is estimated using a neural network. Meanwhile an observer is designed to determine the orbit of Lagrangian satellite. We reference the design of a reduced-order modified state observer which is introduced in [17]. However, in [17], the authors assume that the position can be measured. In our study, we improve the observer which can estimate the state of Lagrangian satellite with only satellite-to-satellite range.

First the dynamical model of ERTBP with perturbation is introduced. Then we design a neural network based state observer to determine the orbit of Lagrangian satellites. Afterwards the stability of the observer is proved. Finally four Lagrangian satellites are chosen to validate the effectiveness of this AOD method.

#### 2. Elliptic Restricted Three-Body Problem with Perturbation

As shown in Figure 1, and are the primaries in the three-body system. and are in elliptical orbits. is the third body which is vanishingly small compared to the two primaries. Similar to the CRTBP, we study the motion of in a rotating coordinates.