Citation: Su, H.; Dong, Z.; Liu, L.;
Xia, L. Numerical Solution for the
Single-Impulse Flyby Co-Orbital
Spacecraft Problem. Aerospace 2022,
9, 374. https://doi.org/10.3390/
aerospace9070374
Academic Editors: Mikhail
Ovchinnikov and Dmitry Roldugin
Received: 24 May 2022
Accepted: 8 July 2022
Published: 11 July 2022
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Article
Numerical Solution for the Single-Impulse Flyby Co-Orbital
Spacecraft Problem
Haoxiang Su, Zhenghong Dong *, Lihao Liu and Lurui Xia
Graduate School, Space Engineering University, Beijing 101416, China; lelouchzero1221@163.com (H.S.);
liulihao070204@163.com (L.L.); xlrui522@163.com (L.X.)
* Correspondence: lelouchzero@buaa.edu.cn; Tel.: +86-13581866513
Abstract:
The traversal inspection of satellites in satellite constellations or geosynchronous orbits
has been a focus of research. A large number of variable orbit requirements in the “single-to-single”
mode severely affects the efficiency of inspections. To address this problem, this study investigated
the problem of a single-impulse flyby co-orbiting two spacecraft and proposed a derivative-free
numerical solution method that used the geometric relationship between the two intersections of the
target and transfer orbits of the flyby problem in order to transform them into a nonlinear equation
in a single variable for a given impulse time. The validity of the proposed method was verified
using numerical examples. While the Lambert problem is one of the bases for solving the variable
orbit problem, on-star intelligent control also raises the requirements for speed. To address this
problem, this study also investigated the Lambert problem in a single-impulse flyby co-orbiting
two spacecraft and determined the iterative initial value by constructing a quadratic interpolation
equation between the inverse of the transfer time and the vertical component of the eccentric vector,
the derivative-free quadratic interpolation cut-off method was proposed. Using 100,000 random tests
showed that computational efficiency was improved by more than one order of magnitude compared
with commonly used methods, with a calculation error of less than 10
−6
.
Keywords: numerical solution; flyby multi-target; Lambert problem
1. Introduction
Maintaining, detecting, or intercepting targets in space has become a vital area of
space technology research, providing countries with a more significant space information
edge. Meanwhile, satellite constellations, which are made up of a large number of satellites
orbiting in the same orbit [
1
,
2
], such as the GPS Navigation System [
3
] and the Beidou Nav-
igation System [
4
,
5
], or satellite communication systems, such as OneWeb and StarLink [
6
],
are playing an indispensable role in society, as well as in the field of national defense.
Therefore, detecting or maintaining the satellites in these constellations has emerged as
an essential research topic. The flyby multi-target problem [
7
,
8
], particularly the flyby
non-coplanar multi-target problem [
9
,
10
], was investigated to some extent, but most so-
lutions require numerous orbit maneuvers and are incredibly dependent on the ground
station [
11
]. Minimizing the number of orbital maneuvers can effectively decrease the
mission constraint, and thus, enhance the efficacy of each orbital maneuver; therefore, it is
crucial to study the single-impulse flyby co-orbital multi-target problem.
The Lambert problem, defined as the problem of finding the impulse time and
value with two given positions and the transfer time, is the fundamental problem in
the
single-target
flyby/interception problem. A typical way of solving the Lambert prob-
lem is to establish a connection between the transfer time and a Kepler element [
12
–
14
]. It
is also common to convert the Lambert problem into an optimization problem by adding
constraints [
15
–
17
] to achieve the optimal solution to the interception problem [
18
,
19
].
Unfortunately, these methods are sensitive to an initial value for iteration, and thus, the
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