`MultiFilterOD`#

class MultiFilterOD(filters: list, global_param_names: List[str], global_apriori_cov: ndarray | None = None, subtract_arc_priors: bool = True)#

Bases: object

Multi-arc orbit determination combining N independent FilterOD instances.

Each arc has its own propagator, measurements, settings, and filter type (SRIF, LKF, LSB, SRIFB, or any FilterOD subclass). The class runs all arcs to convergence and then fuses their estimates of any parameters designated as global (shared across arcs) using the NAPEOS partitioned normal-equation formulation with Schur-complement reduction.

Algorithm overview

The full state vector of arc \(i\) is partitioned into arc-local parameters \(\mathbf{a}_i\) and global (shared) parameters \(\mathbf{g}\):

\[\begin{split}\mathbf{x}_i = \begin{bmatrix} \mathbf{a}_i \\ \mathbf{g} \end{bmatrix}\end{split}\]

Step 1 — Per-arc convergence. Each arc filter is run to inner convergence independently, yielding a state deviation \(\delta\mathbf{x}_{0,i}\) and covariance \(P_{0,i}\) referred to a common reference epoch \(t_0\).

For batch or smoother outputs the solution is already at \(t_0\). For sequential filters (LKF, SRIF) the final-epoch result is mapped back via the cumulative state-transition matrix \(\Phi_N\):

\[\delta\mathbf{x}_{0,i} = \Phi_N\,\delta\mathbf{x}_{N,i}, \qquad P_{0,i} = \Phi_N\,P_{N,i}\,\Phi_N^T\]

Step 2 — Normal equations. The information matrix and right-hand side for arc \(i\) are

\[N_i = P_{0,i}^{-1}, \qquad \mathbf{b}_i = N_i\,\delta\mathbf{x}_{0,i}\]

When subtract_arc_priors=True the arc’s own apriori information is removed so it is counted only once via global_apriori_cov:

\[N_i \leftarrow N_i - P_{\mathrm{apr},i}^{-1}\]

Step 3 — Partitioning and Schur complement. The normal matrix is split into local and global blocks:

\[\begin{split}N_i = \begin{bmatrix} N_{a,i} & N_{ag,i} \\ N_{ag,i}^T & N_{g,i} \end{bmatrix}, \qquad \mathbf{b}_i = \begin{bmatrix} \mathbf{b}_{a,i} \\ \mathbf{b}_{g,i} \end{bmatrix}\end{split}\]

Local parameters are eliminated via the Schur complement:

\[ \begin{align}\begin{aligned}\tilde{N}_{g,i} = N_{g,i} - N_{ag,i}^T N_{a,i}^{-1} N_{ag,i}\\\tilde{\mathbf{b}}_{g,i} = \mathbf{b}_{g,i} - N_{ag,i}^T N_{a,i}^{-1} \mathbf{b}_{a,i}\end{aligned}\end{align} \]

Step 4 — Global combination. An arbitrary reference \(\bar{\mathbf{g}}\) (arc-0 absolute estimate) is used to form the combined information and normal vector:

\[ \begin{align}\begin{aligned}\Lambda = P_{\mathrm{prior}}^{-1} + \sum_{i=1}^{N} \tilde{N}_{g,i}\\\hat{\mathbf{n}} = \sum_{i=1}^{N} \Bigl( \tilde{\mathbf{b}}_{g,i} + \tilde{N}_{g,i}\, (\mathbf{x}_{G,i} - \bar{\mathbf{g}}) \Bigr)\end{aligned}\end{align} \]

giving the combined global estimate and covariance:

\[\hat{P}_G = \Lambda^{-1}, \qquad \hat{\mathbf{x}}_G = \bar{\mathbf{g}} + \hat{P}_G\,\hat{\mathbf{n}}\]

The solution is insensitive to the choice of \(\bar{\mathbf{g}}\).

Step 5 — Local back-substitution. With \(\hat{P}_G\) known, the arc-local correction is recovered by a second Schur inversion:

\[ \begin{align}\begin{aligned}\hat{P}_{a,i} = \bigl( N_{a,i} - N_{ag,i}\,\hat{P}_G\,N_{ag,i}^T \bigr)^{-1}\\\Delta\mathbf{x}_{a,i} = \hat{P}_{a,i}\, \bigl( \mathbf{b}_{a,i} - N_{ag,i}\,\hat{P}_G\,\hat{\mathbf{n}} \bigr)\end{aligned}\end{align} \]

Step 6 — Reinitialization. Each arc is restarted from the corrected reference trajectory \(\mathbf{x}_{\mathrm{ref}} + [\Delta\mathbf{x}_{a,i};\,\Delta\mathbf{x}_G]\) and the outer loop repeats until the relative RMS change in \(\hat{\mathbf{x}}_G\) falls below convergence_threshold.

Parameters:

filters (list of FilterOD) – One filter per arc, fully configured (propagator, measurements, settings). All filters must estimate the same set of global parameters (matching base names) plus their own arc-local parameters.
global_param_names (list of str) – Base names of parameters shared across arcs (e.g. ["gm", "cnm_2_0"]). Every arc must estimate at least one such parameter.
global_apriori_cov (np.ndarray of shape (n_G, n_G), optional) – Apriori covariance for the global parameter vector. Enters as \(P_{\mathrm{prior}}^{-1}\) in the global combination step. Set to None (default) to omit (improper prior).
subtract_arc_priors (bool, optional) – If True (default), each arc’s own apriori covariance \(P_{\mathrm{apr},i}\) is subtracted from \(N_i\) before combination, preventing the prior from being counted once per arc. Set to False to retain the classic behaviour in which every arc contributes its full posterior information matrix including the prior.

Raises:

ValueError – If any arc filter has flag_sequence=True (multi-leg arcs are not supported).
ValueError – If any arc filter estimates no parameter whose name appears in global_param_names.

Notes

Multi-leg (sequence) arcs are not supported; each arc must be a single-leg filter.
All arcs must estimate the same set of global parameters (matched by base name); the dimension \(n_G\) must be consistent across arcs.
The outer loop reinitializes arcs after each global combination; the number of outer iterations required is typically 2–4 for well-posed problems.
If no global_apriori_cov is supplied, the combined problem has an improper prior and the solution is determined solely by the arc data.

MultiFilterOD#

`MultiFilterOD`#