International Journal of Aerospace Engineering

Volume 2017, Article ID 5451908, 10 pages

https://doi.org/10.1155/2017/5451908

## Time and Covariance Threshold Triggered Optimal Uncooperative Rendezvous Using Angles-Only Navigation

^{1}The College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China^{2}Swiss Space Center, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland^{3}Signal Processing Lab (LTS5), École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland^{4}The Manned Space Technology System Center, Beijing 100094, China

Correspondence should be addressed to Yue You; moc.621@tdun_euyuoy

Received 2 April 2016; Accepted 13 November 2016; Published 24 January 2017

Academic Editor: Paul Williams

Copyright © 2017 Yue You 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

A time and covariance threshold triggered optimal maneuver planning method is proposed for orbital rendezvous using angles-only navigation (AON). In the context of Yamanaka-Ankersen orbital relative motion equations, the square root unscented Kalman filter (SRUKF) AON algorithm is developed to compute the relative state estimations from a low-volume/mass, power saving, and low-cost optical/infrared camera’s observations. Multi-impulsive Hill guidance law is employed in closed-loop linear covariance analysis model, based on which the quantitative relative position robustness and relative velocity robustness index are defined. By balancing fuel consumption, relative position robustness, and relative velocity robustness, we developed a time and covariance threshold triggered two-level optimal maneuver planning method, showing how these results correlate to past methods and missions and how they could potentially influence future ones. Numerical simulation proved that it is feasible to control the spacecraft with a two-line element- (TLE-) level uncertain, 34.6% of range, initial relative state to a 100 m v-bar relative station keeping point, at where the trajectory dispersion reduces to 3.5% of range, under a 30% data gap per revolution on account of the eclipse. Comparing with the traditional time triggered maneuver planning method, the final relative position accuracy is improved by one order and the relative trajectory robustness and collision probability are obviously improved and reduced, respectively.

#### 1. Introduction

There are hundreds of thousands of debris in Earth’s orbit; while the biggest are tracked through ground radar, the large majority of pieces remain invisible. The 2009 collision between Iridium 33 and Kosmos-2251 destroyed an active satellite worth tens of millions of dollars [1]. There is international recognition that 5–10 pieces of the currently existing large debris should be removed each year by 2020. ESA’s CleanSpace program [2] is targeted at developing these capabilities, with the specific target of EnviSat, an 8-ton satellite that poses a high risk of a catastrophic collision. CleanSpace One (CSO) is EPFL’s hugely ambitious response to the problem of orbital debris [3]. The most capable nanosatellite ever built, CSO will autonomously intercept, capture, and deorbit a targeted object, EPFL’s own “Swiss Cube,” a 1 kg satellite launched in 2009. The disruptive technologies developed and demonstrated through this project will form the baseline for the upcoming active debris removal (ADR) efforts in Switzerland and abroad. The technologies will also serve for other uncooperative rendezvous mission, such as on-orbit servicing and inspection mission, potentially leading to the creation of autonomous orbital inspection and repair drones.

The practice of single-optical/infrared camera based on AON provides a low-volume/mass, power saving, and low-cost solution for the nanosatellite in ADR. For these reasons, autonomous vehicles using AON are currently an active area of research [4–14]. The growing interest in vision-based autonomous rendezvous and docking has produced a series of experimental spacecraft in an attempt to develop rendezvous and proximity operations technology that would be more appropriate for small satellite. PRISMA [15] is the only in-orbit test bed of angles-only navigation, OHB Sweden [16], CNES [17], and DLR [18] imitated uncooperative rendezvous using AON separately, but the uncertainty is accuracy known with GPS and RF. Besides, all the rendezvous processes are ground-in-loop control; real AON based autonomous uncooperative rendezvous needs to be further studied.

A fundamental challenge faced by AON is the inherent difficulty in accurately determining the range to target [7]. In rigorous terms, the navigation problem becomes observable only in the presence of orbit control maneuvers, which change the relative orbit geometry of the observation [11]. Woffinden and Geller [6, 7] and Grzymisch and Fichter [14] separately derived an observability criterion for AON and optimized a single-pulse maneuver to minimize estimation error in relative range direction. Spurmann [19] proposed the spiraling maneuver approach of AON to cover the transition gap from absolute to relative navigation. Jaehwan and Hyochoong [20] implemented the observability constraint at each rendezvous maneuver optimization node in which the cost function was aimed at minimizing fuel consumption; their method minimized fuel consumption while ensuring sufficient relative observability during the whole rendezvous maneuver. All the previous researches made observability as an optimization cost function or constraint, being offline and open loop [21]. But for uncooperative rendezvous especially long-time autonomous mission, the modeling error and control error exists, the adjacent maneuvers interval constraints, and data gap induced by solar eclipse must be taken into concern as well [22]. Besides observability, safety which is related to trajectory dispersion and fuel efficiency since satellites usually have a tight mass budget should be tradeoff at the meantime. Tang et al. [23] and Luo et al. [24–26] took the navigation and control error into consideration and defined the open-loop optimal robust rendezvous planning method, making tradeoff between fuel consumption, rendezvous time, and trajectory robustness. Li et al. [27] furtherly constructed closed-loop multiple objective optimization problem (MOOP) considering the position robustness, velocity robustness, and fuel. Both the previous maneuver planning methods employed the heavy computational burden physical planning and nondominated sorting genetic algorithm (NSGA-II), which is not applicable for onboard considering limit computational resource.

In this study, the relative rendezvous trajectory is optimized for AON between chaser (active satellite or space robot) and target (uncooperative satellite or space debris). The main purpose of this paper is to design an online rendezvous maneuver planning method which provides better observability and robustness for the whole mission period. The second purpose is to improve fuel consumption efficiency. There is a tradeoff between the relative position robustness, relative velocity robustness, and fuel consumption.

The remainder of the paper is organized as follows. Problem Formulation defines the problem by addressing the basics of relative dynamic and optical navigation. Closed-Loop Linear Covariance Analysis introduces the analytical linear covariance method to quantify the robustness measure for closed-loop relative trajectory dispersion. In Optimal Maneuver Planning, a rendezvous maneuver design strategy is proposed to minimize fuel consumption while seeking high position and velocity robustness. Using the proposed strategy, a numerical optimization technique is implemented to design the rendezvous trajectory. The designed trajectory is verified by performing Monte Carlo simulation. Conclusion presents concluding remarks for this study.

#### 2. Problem Formulation

##### 2.1. Dynamic Modeling

The origin of a rotating LVLH (Local Vertical Local Horizontal) reference frame (), which is used for relative motion description, is collocated with the debris c.m. (c.m. is short for center of mass). The relative position and velocity of the chaser c.m. with respect to the target c.m. in the LVLH coordinates are denoted by and , respectively. The relative motion equations for general elliptical orbits are the well-known Tschauner-Hempel (TH) equations [28] whose homogeneous solution is known as the Yamanaka-Ankersen state transition matrix [29]where the respective expressions of and are as follows:with , , , , and for shorthand notation. In addition, is the eccentricity of the target orbit, is the true anomaly, is the gravitational constant of Earth, and is the norm of target’s angular momentum. Here, and indicate the first derivatives with respect to , respectively.

These solutions can be written in discrete form for impulsive input aswhere and are chaser’s relative states at initial moment and time , respectively. is the Dirac function.

##### 2.2. Camera Observation Modeling

The camera measurement frame () is assumed to be aligned with the focal plane of the camera. Its orientation with respect to the chaser body frame is assumed to be known and constant. The pixel location of the debris c.m. is used to form an line of sight (LOS) vector from the debris c.m. to the camera, which is expressed in the camera frame and denoted by , as shown in Figure 1. Because the transformation from LVLH to the camera measurement frame can be calculated using knowledge of inertial attitude, position, and velocity, the LOS measurement expressed in the LVLH frame can be used to formulate the angle/bearing-only measurement equation providing the measurement angles to the debris, which is explicitly written as follows:where and are the respective elevation and azimuth measurement angles. is the average radius of the debris circular motion.