Optimization Methods

The shooting method converts the orbital insertion problem into a nonlinear least-squares problem.

LaTeX Reference

This section corresponds to Section 8 of nm_final_project.tex.

The Challenge¶

We want to find control parameters that achieve a circular orbit:

\[ \text{Find } \mathbf{u} = [\theta_0, t_{\text{coast}}, t_{\text{burn2}}, \alpha_2]^T \]

Such that the final orbit is:

At target altitude: \(a = R_E + h_{\text{target}}\)
Circular: \(e = 0\)
Horizontal insertion: \(\gamma = 0\)

Topics¶

Shooting Method - Problem formulation
Levenberg-Marquardt - Optimization algorithm
Broyden Update - Jacobian acceleration

Algorithm Overview¶

graph TD
    A[Initial Guess u₀] --> B[Simulate Trajectory]
    B --> C[Compute Orbital Elements]
    C --> D[Evaluate Residuals F(u)]
    D --> E{||F|| < tol?}
    E -->|Yes| F[Converged!]
    E -->|No| G[Compute Jacobian J]
    G --> H[LM Step: Δu]
    H --> I[Line Search]
    I --> J[u ← u + α·Δu]
    J --> K[Broyden Update J]
    K --> B

Mathematical Formulation¶

See Shooting Method for the complete derivation of:

\[ \mathbf{F}(\mathbf{u}) = \begin{bmatrix} \frac{a(\mathbf{u}) - r_{\text{target}}}{r_{\text{target}}} \\ e(\mathbf{u}) \\ \gamma(\mathbf{u}) \end{bmatrix} \approx \mathbf{0} \]