Abstract
This paper studies the relative motion of satellite formation flying in arbitrary elliptical orbits with no perturbation. The trajectories of the leader and follower satellites are projected onto the celestial sphere. These two projections and celestial equator intersect each other to form a spherical triangle, in which the vertex angles and arc-distances are used to describe the relative motion equations. This method is entitled the reference orbital element approach. Here the dimensionless distance is defined as the ratio of the maximal distance between the leader and follower satellites to the semi-major axis of the leader satellite. In close formations, this dimensionless distance, as well as some vertex angles and arc-distances of this spherical triangle, and the orbital element differences are small quantities. A series of order-of-magnitude analyses about these quantities are conducted. Consequently, the relative motion equations are approximated by expansions truncated to the second order, i.e. square of the dimensionless distance. In order to study the problem of periodicity of relative motion, the semi-major axis of the follower is expanded as Taylor series around that of the leader, by regarding relative position and velocity as small quantities. Using this expansion, it is proved that the periodicity condition derived from Lawden’s equations is equivalent to the condition that the Taylor series of order one is zero. The first-order relative motion equations, simplified from the second-order ones, possess the same forms as the periodic solutions of Lawden’s equations. It is presented that the latter are further first-order approximations to the former; and moreover, compared with the latter more suitable to research spacecraft rendezvous and docking, the former are more suitable to research relative orbit configurations. The first-order relative motion equations are expanded as trigonometric series with eccentric anomaly as the angle variable. Except the terms of order one, the trigonometric series’ amplitudes are geometric series, and corresponding phases are constant both in the radial and in-track directions. When the trajectory of the in-plane relative motion is similar to an ellipse, a method to seek this ellipse is presented. The advantage of this method is shown by an example.
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Abbreviations
- (A, B):
-
semi-major and semi-minor axes of the ellipse approximate to the in-plane relative motion
- a :
-
semi-major axis of the leader satellite
- d j :
-
the j-th integration constants in Lawden’s equations
- E :
-
eccentric anomaly of the leader satellite
- e :
-
eccentricity of the leader satellite
- (err x , err y ):
-
maximum errors between truncated elliptical motion and accurate relative motion in the x- and y-axes
- (erx, ery):
-
indexes to evaluate (err x , err y )
- F :
-
coefficient with respect to amplitudes of trigonometric series
- f :
-
true anomaly of the leader satellite
- (G j , H j ):
-
the j-th coefficients of the trigonometric series of x 2/a 2
- i :
-
orbit inclination of the leader satellite
- i, j, k :
-
unit vector in the X-, Y- and Z-axes of Earth-centered-inertial frame
- i c , j c , k c :
-
unit vector in the x-, y- and z-axes of the leader’s LVLH frame
- (J j , K j ):
-
the j-th coefficients of the trigonometric series of y 2/a 2
- l r :
-
reference mean ascension
- M :
-
mean anomaly of the leader satellite
- O(10−k):
-
a value whose order of magnitude is not larger than 10−k
- (P j , Q j ):
-
the j-th coefficients of the trigonometric series of xy/a 2
- (R j , S j ):
-
the j-th coefficients of the trigonometric series of (x 2 + y 2)/a 2
- r :
-
position vector from the Earth center to the leader satellite
- r :
-
magnitude of r
- v :
-
velocity vector of the leader satellite with respect to the Earth
- v :
-
magnitude of v
- (X j , U j ):
-
the j-th coefficients of the trigonometric series of x/a
- (x, y, z):
-
radial, in-track and cross-track position distances from the origin of the LVLH frame
- \({\left({\hat{x}, \hat{y}}\right)}\) :
-
approximate elliptical motion equations in the x − y plane
- (x 0, y 0):
-
center coordinates of the approximate ellipse
- (Y j , V j ):
-
the j-th coefficients of the trigonometric series of y/a
- α:
-
angle parameter
- Δ β:
-
difference angle parameter
- \({\varepsilon}\) :
-
index function to weigh the similarity between approximate elliptical motion and the first-order relative motion
- \({(\varepsilon_{j},\upsilon_{j})}\) :
-
the j-th coefficients of the trigonometric series of \({\varepsilon}\)
- η:
-
spherical angle with respect to the leader satellite
- θ:
-
argument of latitude of the leader satellite
- \({\vartheta }\) :
-
included angle between the semi-major axis of the approximate ellipse and the x-axis
- λ:
-
common ratio, a function of e
- μ:
-
gravitational parameter of the Earth, 3.986005 × 1014 m3/s2
- χ(E):
-
phase function of the approximate elliptical motion in the x–y plane
- Ω:
-
right ascension of ascending node of the leader satellite
- ω:
-
argument of perigee of the leader satellite
- d(·):
-
differential of (·)
- Δ (·):
-
difference of (·) between the leader and follower satellites
- \({\nabla (\cdot)}\) :
-
gradient of (·)
- \({\delta (\cdot)}\) :
-
variation of (·)
- \({\mathop {\mathop {\left( \cdot \right)}\limits^\cdot }\limits_\sim }\) :
-
derivative of (·) with respect to time in inertial frame
- \({\mathop {\left( \cdot \right)}\limits^\cdot}\) :
-
derivative of (·) with respect to time in LVLH frame
- (·)′:
-
derivative of (·) with respect to f
- f :
-
value of the follower satellite
- r :
-
reference value of the follower with respect to the leader
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Jiang, F., Li, J. & Baoyin, H. Approximate analysis for relative motion of satellite formation flying in elliptical orbits. Celestial Mech Dyn Astr 98, 31–66 (2007). https://doi.org/10.1007/s10569-007-9067-8
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DOI: https://doi.org/10.1007/s10569-007-9067-8