Formation Flight

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

Formation flight is a mission mode in which multiple spacecraft fly cooperatively while maintaining a specific relative configuration. In the three-body environment, relative motion is analyzed with the linearized relative-motion equations plus Floquet decomposition: the bounded solution set already contains periodic and quasi-periodic solutions, and only the divergent ones need to be removed — boundedness is the essential requirement for preventing drift and collision between members (Yang, Wang & Zhang 2023). Compared with two-body orbits, the strong nonlinearity of three-body dynamics makes formation keeping more dependent on the choice of dynamical structure.

Triangular Libration Point Formations

Catlin & McLaughlin 2007 derived the relative motion near Earth–Moon L4 in the CR3BP, decomposing it into long-period and short-period components that are designed separately (when both periods coexist the relative motion is a complex curve with axis ratio about 16/5 and period 458 days, unsuitable for formations):

• Parallel formation: members keep a fixed in-plane phase offset while their out-of-plane components are identical at all times. The most tolerant of injection errors (under the maintenance constraint, an initial-value accuracy of 1000% — i.e. 26 km and 1 m/s — suffices), it is the most robust natural formation in the paper.
• Leader–follower formation: members trail one another along the same trajectory.
• Circular formation: the long-period motion has an ellipse axis ratio of about 16/3 (period about 92 days), so a natural circular formation is impossible; the short-period motion (axis ratio about 2, period about one month) yields, by planar approximation, an approximate circle inclined about 60°, but it requires active control to maintain (an impulse reset at the end of each revolution, or continuous correction).

Background and value: no spacecraft had yet reached the triangular libration points (as of 2007), and getting there costs significantly more than reaching the collinear points; the value lies in unobstructed lines of sight for deep-space observation and radiation monitoring, and in future space-station siting. Perturbation ranking: solar point-mass gravity most significant, solar radiation pressure next, Earth's J2 negligible (Catlin 2007).

Relative-Motion Modes on DROs

For a reference DRO (13.64 days, 2:1 resonance), the bounded solutions of the linearized relative motion combine into three modes (Yang, Wang & Zhang 2023):

• Planar periodic mode (also called natural periodic mode): the deputy sits on the reference DRO with a phase offset; the trajectory stays on one side of the chief only, suiting it as a parking orbit for rendezvous and docking.
• Planar quasi-periodic mode: composed of two periodic components, the trajectory rotates around the chief within a bounded hollow region — the basic mode for fly-around formations. Note that the geometry of purely natural long-term fly-around is irregular; the engineering fly-around is a patchwork of "a natural segment within one DRO period + a transfer segment," needing only two impulses per revolution and less than 1 cm/s of fuel per revolution for a 1 km-scale formation.
• Normal quasi-periodic mode: contains only the normal (z) component and serves to extend a planar formation into three dimensions — it can only be superimposed on a planar formation, never forms a formation alone.

DRO Formation Design

Yang et al. 2023 (Chinese Journal of Aeronautics, Chinese-language edition — a sister paper by the same team as the English paper above) give two designs:

• Natural accompanying formation: based on the periodic solutions among the bounded ones, the deputy accompanies the chief a few meters to several hundred kilometers ahead or behind, long-term and low-fuel; but the natural configurations are very limited and the relative motion is slow.
• Circular controlled fly-around formation: with a spatial circle as the reference trajectory, impulses are applied at evenly spaced maneuver points to control the deputy's fly-around (the example uses 10 maneuver points per revolution, at scales of 1–100 km); the fuel cost depends on the circle's normal direction, center position, and scale — this is impulsive control, not continuous thrust.

Phase Difference and Rendezvous

The angular spacing along the orbit between two spacecraft on the same halo orbit is called the phase difference. In same-orbit rendezvous, for equal transfer time the larger the initial phase difference, the larger the required velocity increment (Sun 2017's example: on an L2 northern halo orbit, a 1° phase difference allows a 3.46-day transfer at only 3.40 m/s, while 5° takes 7.35 days at 49.85 m/s); for rendezvous between halo orbits of different amplitudes, the larger the amplitude difference, the larger the ΔV. So whenever safety permits, the phase difference should be as small as possible.

Terminology Variants

Related Concepts

• Distant Retrograde Orbit (DRO)
• Multi-Body Constellation
• Halo Orbit
• Quasi-Periodic Orbit (QPO)

References

• Catlin & McLaughlin, 2007, Earth–Moon triangular libration point spacecraft formations
• Sun, Zhang & Luo, 2017, Libration-point rendezvous trajectory design based on a three-body Lambert algorithm
• Yang, Wang & Zhang, 2023, Close relative motion on distant retrograde orbits
• Yang, Fu & Zhang, 2023, Close-proximity natural and controlled formation flying on distant retrograde orbits (Chinese Journal of Aeronautics 44(5):326563, in Chinese)