Implements the shooting method with Levenberg-Marquardt optimization.

Functions¶

compute_residuals¶

Evaluate the shooting residual function.

def compute_residuals(
    u: np.ndarray, 
    base_config: AscentConfig
) -> tuple[np.ndarray, Trajectory]:

Control vector \(\mathbf{u}\): 1. \(\theta_0\) - Pitchover angle [rad] 2. \(t_{\text{coast}}\) - Coast duration [s] 3. \(t_{\text{burn2}}\) - Stage 2 burn time [s] 4. \(\alpha_2\) - Stage 2 steering [rad]

Residual vector \(\mathbf{F}\) (Eq. 22): 1. \((a - r_{\text{target}}) / r_{\text{target}}\) 2. \(e\) (eccentricity) 3. \(\gamma\) (flight-path angle)

solve_circular_orbit¶

Main optimization function.

def solve_circular_orbit(
    base_config: AscentConfig
) -> tuple[np.ndarray, Trajectory]:

Algorithm:

1. Multistart candidate generation
2. Newton iteration with LM damping
3. Line search
4. Broyden Jacobian updates

Returns: (optimal_u, trajectory)

Internal Functions¶

_u_from_x / _x_from_u¶

Logistic transform for bounded optimization (Eq. 23).

_fd_jacobian_x¶

Finite-difference Jacobian approximation (Eq. 25).

_broyden_update¶

Rank-1 Jacobian update (Eq. 26).

_newton¶

Main Newton iteration with LM and line search.

Equation Reference¶