Shooting Method

Author: 天疆说

Site: https://cislunarspace.cn

Definition

The Shooting Method is a numerical technique that transforms a Two-Point Boundary Value Problem (TPBVP) into an initial value problem solved by iterative correction. Its core idea is: guess the initial state, integrate to the terminal constraint, compute the deviation, correct the initial guess, and iterate until convergence.

Principles

In orbital mechanics, the shooting method is commonly used to generate periodic orbits. Taking Halo orbits as an example, the procedure is:

1. Select an initial guess $\mathbf{x}_0 = (x_0, 0, z_0, 0, \dot{y}_0, 0)$ on a reference manifold (e.g., the $xOz$ plane)
2. Integrate until the orbit crosses the periodic constraint surface (e.g., the $xOz$ plane again)
3. Compute the state deviation $\Delta \mathbf{x}_f = \mathbf{x}_f - \mathbf{x}_0$
4. Linearize using the State Transition Matrix (STM) $\mathbf{\Phi}$: $\Delta \mathbf{x}_f = \mathbf{\Phi} \cdot \Delta \mathbf{x}_0$
5. Correct the initial guess iteratively until the periodic condition is satisfied

Deviation Correction

The key in shooting is selecting the shooting variables and targeting equations. For Halo orbits, $z_0$ and $\dot{y}_0$ are typically chosen as shooting variables, with targeting equations $y=0$ and $z=0$ at the crossing surface.

Applications in Cislunar Space

The shooting method is widely used in cislunar orbit design:

• NRHO initial condition generation: Zimovan (2017) systematically summarized single-shooting and multi-shooting strategies for Earth-Moon L1/L2 NRHO
• DRO orbit generation: Exploiting $x$-axis symmetry, initial points are selected on the $x$-axis only, with $\dot{y}_0$ and period $T$ as shooting variables
• Halo orbit family continuation: Starting from planar Lyapunov orbits, arc-length continuation gradually increases $A_z$ amplitude, with shooting at each step

The shooting method is typically combined with arc-length continuation and differential correction to improve convergence and global coverage.

Related Concepts

• Arc-length Continuation
• Symplectic Integrator
• Distant Retrograde Orbit (DRO)
• Near-Rectilinear Halo Orbit (NRHO)
• Circular Restricted Three-Body Problem (CR3BP)

References

• Zimovan E M. Characteristics and design strategies for near rectilinear halo orbits within the Earth-Moon system[D]. Purdue University, 2017.
• Liu X, Baoyin H, Ma X. Design of optimal lunar landing trajectories with plane change[J]. Advances in Space Research, 2022.