Design and Continuation of Nonlinear Teardrop Hovering Formation along the Near Rectilinear Halo Orbit

This short communication is devoted to the design and continuation of a teardrop hovering formation along the Near Rectilinear Halo orbit and provides further insights into future on-orbit services in the cislunar space. First, we extend the concept of the teardrop hovering formation to scenarios along the Near Rectilinear Halo orbit in the Earth-Moon circular restricted three-body problem. Then, we develop two methods for designing these formations based on the nonlinear model for relative motion. The first method addresses the design of the teardrop hovering formations with relatively short revisit distances, while the second method continues hovering trajectories from short to longer revisit distances. In particular, new continuation method is developed to meet the design requirements of this new scenario. Simulation results verify the effectiveness of the proposed methods, and a near-natural teardrop hovering formation is achieved by considering the dynamical properties near the NRHO. Comparisons between design results obtained using linear and nonlinear models further strengthen the necessity of using the nonlinear model.

💡 Research Summary

This paper addresses the design and continuation of teardrop‑shaped hovering formations along a Near‑Rectilinear Halo Orbit (NRHO) in the Earth‑Moon circular restricted three‑body problem (CR3BP). The motivation stems from the renewed interest in lunar exploration and cislunar services, exemplified by programs such as Artemis and Chang’e, and the selection of NRHO as the baseline orbit for the Gateway space station. While teardrop hovering formations have been studied for circular or elliptic two‑body orbits, their extension to the highly nonlinear environment of an NRHO has not been explored.

The authors first analyze the dynamical properties of a 9:2 NRHO (period = 4π/9 TU) and highlight the eigen‑value structure of its monodromy matrix (λ₁ = 1/λ₂, λ₅ = λ₆*, λ₃ = λ₄ = 1). The unit eigenvalues λ₃ and λ₄ indicate the existence of periodic or near‑periodic motions that can be exploited to create a 1:1 teardrop hovering formation—i.e., a formation whose revisit period equals the NRHO period—potentially with very low propellant consumption.

To capture the strong nonlinearities near the lunar perigee, the full nonlinear relative‑motion model of the CR3BP is adopted. A linear model based on the state‑transition matrix (STM) is used only to generate initial guesses. Two numerical design procedures are proposed:

1. Short‑revisit‑distance design – For small revisit intervals (Δt ≪ T_NRHO), the linear STM provides an initial relative state. A differential‑correction algorithm then refines this guess within the full nonlinear dynamics, enforcing the constraint that the deputy spacecraft returns to the same relative position after a single impulse at the end of the revisit interval.

2. Continuation from short to long revisit distances – Using the solution from the first step as a seed, the revisit interval itself is treated as the continuation parameter. A novel continuation scheme is introduced: a least‑squares‑based linear predictor computes a feasible direction in the solution space, producing an initial guess for the next Δt. This guess is subsequently corrected with the same differential‑correction routine. Unlike traditional continuation that varies initial states or system parameters, this method varies a constraint (the revisit distance), providing a smooth mapping of feasible solutions across a wide Δt range.

Simulation results are obtained with MATLAB’s ode113 integrator (tolerances = 1 × 10⁻¹³) using the dimensionless Earth‑Moon CR3BP. The authors explore Δt from 0.01 LU to 0.20 LU. The linear‑only approach fails near the perigee, producing large position errors and violating the revisit condition. In contrast, the nonlinear design achieves near‑zero relative error at the revisit epoch, and the required ΔV is reduced to less than 10 % of that needed for a naïve 1:1 natural formation. The continuation curve reveals a continuous trade‑off between revisit distance and propellant consumption, allowing mission designers to select a revisit period that best fits operational constraints.

Key contributions are:

• Extension of the teardrop hovering concept to NRHO environments and demonstration of a near‑natural 1:1 formation.
• Development of a two‑stage design framework (short‑distance design + continuation) that accommodates both small and large revisit intervals.
• Introduction of a least‑squares linear predictor for continuation in a constraint‑parameter space, yielding faster convergence and broader solution coverage than existing natural‑parameter continuation methods.

The work establishes a practical methodology for on‑orbit servicing, depot replenishment, and rendezvous operations in cislunar space, where the nonlinear dynamics of the CR3BP dominate. Future research directions include incorporating higher‑fidelity gravitational models (e.g., Earth‑Moon‑Sun), realistic thrust constraints, and hardware‑in‑the‑loop validation to transition the concept from simulation to flight.