Closed-form solution of repeat ground track orbit design and constellation deployment strategy

doi:10.1016/j.actaastro.2020.12.021

Acta Astronautica

Volume 180, March 2021, Pages 588-595

https://doi.org/10.1016/j.actaastro.2020.12.021 Get rights and content

Highlights

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Herein, the potential advantages of RGT orbits for earth observation are explored.
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The aim is to improve the RGT orbit performance based on satellite responsiveness.
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The revisit time was reduced to approximately 0.55–0.60 times that of Walker–Delta.
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Thus, the effectiveness of the proposed design was proved.

Abstract

The present work proposes a closed-form solution for circular repeat ground track (RGT) orbit designs and constellations and its application strategy. The proposed solution is an equation of motion proving that the characteristic of a circular RGT orbit is similar to that of the parametric curve of Ptolemy's model and provides a low computational cost for the RGT orbit design and related numerical analysis. The application strategy is to coincide with the self-intersection point of an RGT orbit over a specific target of interest on Earth and make it the starting point of the first satellite orbit in the satellite group. The remaining satellites are designed to have the same RGT orbit as the first satellite; therefore, the entire group has twice the repeat cycle over a specific target. To this end, this study represents an algorithm to match the specific target and the intersection point through an iterative numerical analysis method centering on the RGT equation. A set of orbital elements for a satellite group is also derived herein. The satellite group with a repeat cycle that doubles for a specific target shows a significantly shorter revisit time compared with the conventional Walker constellation.

Introduction

Current trends in satellite development are increasing the utility of many low-cost micro-satellites, which are replacing small- and medium-sized satellites. Increase of security threats (e.g., facilities for nuclear weapons or weapons of mass destruction in a few hostile countries) has highlighted the importance of a responsive mission for the intensive monitoring of a specific target/area using a large number of low-cost micro-satellites. However, related literature reviews have not been actively studied for the revisit time or coverage issues of a specific target.

From the viewpoint of narrowing the area in the continuous coverage problem, the Walker system was initially aimed at guaranteeing global coverage with a minimal number of satellites [1,2]. Later studies then utilized specific orbits, such as polar orbits, to optimize continuous coverage [3,4]. With the emergence of the Streets-of-Coverage concept, the global coverage problem narrowed the area of interest for satellite constellations [5,6]. On the contrary, compared to the continuous coverage problem, the periodic coverage problem has progressed in the research of a narrow class of coverage [[7], [8], [9], [10], [11]]. The periodic coverage problem is more complex than the continuous coverage problem as it considers the rotation of the Earth. It is represented by a repeat ground track (RGT) orbit, which has an identical viewpoint to the same point on the Earth's surface during the repeat cycle, thus offering many advantages for Earth observation. Several studies have focused on RGT orbit designs that observe the ground target from an adjacent orbit considering the tilt capacity or the swath width of sensors [12,13]. Moreover, satellite constellation techniques based on the ground track characteristic analysis or ground track distance adjustment for the observation of a given area have been studied [14,15].

One related work conducted a study to minimize the revisit time of a small satellite group and maximize the percent coverage using a genetic algorithm for a specific target region defined by a grid point [16]. The study investigated the continuous coverage problem for the specific target region. Recent studies investigated the revisit time for a point with a special type of recursive orbit rather than the perspective of the satellite swath width and maximum revisit time [17,18]. In these studies, the goal is to improve the performance of RGT orbits by visiting a single target at its ascending and descending stages by applying the concept of satellite responsiveness. Consequently, the intersection point of the RGT orbit for one satellite was found using the optimization technique.

The present study discusses a design strategy that minimizes the revisit time using the intersection point of the RGT orbit for the responsive mission of a satellite group to a specific target. Importantly, this study has two academic contributions in RGT orbit dynamics: one is the proposal of a closed-form formula with a low computational cost that simply designs circular RGT orbits and multiple satellites with an identical constellation pattern; and the other is the suggestion of a satellite deployment strategy that minimizes the revisit time by doubling the repeat cycle through matching the RGT orbit intersection point of the satellite group over a specific target. The remainder of this article is organized as follows. In Section 2, the equations of motion for the RGT orbit are defined by transforming the geometrically developed equations into a parametric form in Ptolemy's model. Section 3 presents the characteristics of RGT orbits and a closed-form formula for the satellite constellations. Section 4 introduces the proposed strategy and presents the results of minimizing the revisit time of the satellite group by utilizing the intersection point of the RGT orbits.

Section snippets

Geometrical approach

A purely geometrical approach was employed to construct equations of motion defining the position of a satellite in relation to its orbit as projected onto the Earth's surface. Assuming that the satellite orbit is circular, the resulting equations were expressed in four (a, Ω, i, and M) out of six orbit elements [19], where a is the semi-major axis describing the orbit's size, Ω the ascending node, i the inclination defining the orbit's planar orientation, and M the mean anomaly determining the

Orbit frequency $γ$

The value of $γ$ is a real number; thus, it plays a decisive role in representing the shape of a satellite orbiting the Earth. If $γ$ is an integer, the satellite will have a shape of a simple-closed periodic orbit. If $γ$ is a decimal integer, then the orbit shape becomes increasingly complex. As the $x y$ -components of a satellite orbit are represented by a parametric curve, its shape takes the form of cusps or a transformed shape of the parametric curve. As a rule, the shape of the satellite's orbit

N-satellite constellation using the node

Based on the RGT closed-form formula and design rules described in the previous section, this section presents a strategy for shortening the revisit time using the node advantage. The RGT orbit strategy aims to match the following three conditions: (a) specific target; (b) node; and (c) starting point of RGT orbit. More specifically, the RGT orbit node is matched over a specific target, and the starting point of the first satellite in the satellite group is set. This node is called the node

Conclusions

The equations of motion used herein proved that the circular RGT orbit has the properties of a hypo-type parametric curve in the xy plane and a simple spring-mass system in the z direction. The characteristics of each RGT orbit parameter can be identified through these dynamical properties. The compact form of the equations of motion and the low computational cost were used as an efficient tool for the numerical analysis of the RGT orbit. For a specific target, the problem of minimizing the

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors would like to thank the anonymous people who provided feedback on this paper and helped with proofreading. This study is expected to be well used in the mission design of the microsatellite constellations in progress by the Republic of Korea Air Force.

Soung Sub Lee is an assistant professor at Sejong University.. He graduated from ROK Air Force Academy and received a master's degree from Yonsei University. In 2009, he received a Ph.D. in space engineering from Virginia Tech. His research interests are in the field of dynamics and control of satellite relative motions.

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View full text

Closed-form solution of repeat ground track orbit design and constellation deployment strategy

Highlights

Abstract

Introduction

Section snippets

Geometrical approach

Orbit frequency γ

N-satellite constellation using the node

Conclusions

Funding

Declaration of competing interest

Acknowledgments

Satellite networks for continuous zonal coverage

J. Am. Rocket Soc.

Design of satellite constellations for optimal continuous coverage

IEEE Trans. Aerosp. Electron. Syst. AES

Circular polar constellations providing continuous single or multiple coverage above a specified latitude

J. Astronaut. Sci.

Near-polar satellite constellations for continuous global coverage

J. Spacecraft Rockets

Optimum polar satellite networks for continuous earth coverage

AIAA J.

Optimized polar orbit constellations for redundant earth coverage

J. Astronaut. Sci.

Fundamentals of the route theory for satellite constellation design for earth discontinuous coverage. Part 1: analytic emulation of the earth coverage

Acta Astronaut.

Fundamentals of the route theory for satellite constellation design for earth discontinuous coverage. Part 2: synthesis of satellite orbits and constellations

Acta Astronaut.

Fundamentals of the route theory for satellite constellation design for earth discontinuous coverage. Part 3: low-cost earth observation with minimal satellite swath

Acta Astronaut.

Fundamentals of the route theory for satellite constellation design for earth discontinuous coverage. Part 4: compound satellite structures on orbits with synchronized nodal regression

Acta Astronaut.

Orbit frequency $γ$