Predictor Integrator

This module defines integrators for use in predictor-corrector frameworks. It includes three classes:

• RK45Integrator: Integrator for 2D (polar) dynamics, supporting both one-step RK45 integration and analytic Jacobian matrix computation for use in Kalman Filters.

• RK45Integrator_3D: Same as RK45Integrator, but for full 3D spherical coordinate dynamics using a different Jacobian function.

• Integrator3D: Subclass of the main simulation Integrator, extended to allow Jacobian estimation via finite differences for Kalman filters or linear approximations.

---

def __init__(self, CD, A, m, GM, rho_func):
    ...

• Parameters:

  • CD: Drag coefficient.

  • A: Cross-sectional area of the satellite.

  • m: Satellite mass.

  • GM: Gravitational parameter (G·Mₑ).

  • rho_func: A function that returns atmospheric density given altitude or radius.

---

def step(self, f, x, dt):
    sol = solve_ivp(lambda t, y: f(y),
                    (0, dt), x,
                    rtol=1e-6, atol=1e-9,
                    max_step=dt / 10)
    return sol.y[:, -1]

• Purpose: Performs a single Runge-Kutta 45 (RK45) integration step.

• Parameters:

  • f: Dynamics function taking a state vector x and returning its time derivative.

  • x: Current state vector.

  • dt: Time increment for integration.

• Returns: State vector after time dt.

---

def transition_matrix(self, x, dt):
    ...
    return expm(F_cont * dt)

• Purpose: Computes the continuous-time Jacobian matrix at state x, then converts it to a discrete-time transition matrix using the matrix exponential.

• Returns: Discrete transition matrix Φ = exp(F·dt) using compute_F_analytic() or compute_F_spherical() from ExtendedKalmanFilters.

---

def step(self, f, x0, dt):
    # use the same tolerances & max_step as run_rk45_3d
    sol = solve_ivp(
        Integrator.rhs_spherical,
        (0.0, dt),
        x0,
        method="RK45",
        rtol=1e-7,
        atol=1e-9,
        max_step=1.5
    )
    return sol.y[:, -1]

• Purpose: Integrates the state vector x0 forward by dt seconds using RK45.

• Note: The f parameter is ignored and hardcoded to Integrator.rhs_spherical.

---

def transition_matrix(self, x0, dt, eps=1e-6):
    # finite-difference Jacobian over dt
    n = len(x0)
    M = np.zeros((n, n))
    base = self.step(None, x0, dt)
    for i in range(n):
        xp = x0.copy()
        xp[i] += eps
        pert = self.step(None, xp, dt)
        M[:, i] = (pert - base) / eps
    return M

• Purpose: Numerically approximates the Jacobian matrix ∂f/∂x using finite differences.

• Parameters:

  • x0: State vector at which to evaluate the Jacobian.

  • dt: Integration duration.

  • eps: Perturbation size for finite difference.