Shooting Method

Author: Tianjiang Says

Website: https://cislunarspace.cn

Definition

The shooting method is a numerical technique for solving boundary value problems (BVPs) by converting them into initial value problems (IVPs). In orbital mechanics, it is widely used to compute periodic orbits, transfer trajectories, and other orbits that must satisfy specific boundary conditions.

Single Shooting Method

The basic approach:

  1. Guess initial conditions (e.g., position and velocity at the starting point)
  2. Propagate the trajectory forward using the equations of motion
  3. Evaluate the residual (difference between the final state and the desired boundary condition)
  4. Update the initial conditions using the state transition matrix to reduce the residual
  5. Iterate until convergence

x0(k+1)=x0(k)−Φ−1r(k)\mathbf{x}_0^{(k+1)} = \mathbf{x}_0^{(k)} - \Phi^{-1} \mathbf{r}^{(k)}

Multiple Shooting Method

For long-duration or sensitive orbits, multiple shooting divides the trajectory into segments and applies boundary conditions at each node. This improves convergence and numerical stability.

Applications

ApplicationDescription
Periodic orbit computationFinding closed orbits that return to their starting point
Transfer orbit designComputing trajectories between two specified states
Orbit continuationTracing orbit families as parameters vary
Differential correctionRefining approximate orbits to satisfy constraints

References

  • Chen Yuju. DRO Orbit Design and Control Research for Cislunar Space Situation Awareness[D]. 2024.