`FirstOrderGaussMarkov`#

class FirstOrderGaussMarkov(state_array: StateArray, spacecraft: Spacecraft, base_frame: Frame, default_beta: ndarray | None = None)#

Bases: DynamicModel

Models a first-order exponentially correlated Gauss-Markov acceleration process.

An empirical acceleration state \(\mathbf{a}_{gm}\) is added to the equations of motion and evolves according to exponential decay:

\[\dot{\mathbf{a}}_{gm} = -\mathbf{B}\,\mathbf{a}_{gm}\]

where \(\mathbf{B} = \mathrm{diag}(\beta_x, \beta_y, \beta_z)\) with \(\beta_i = 1/\tau_i\) being the inverse correlation times. The spacecraft translational dynamics become \(\ddot{\mathbf{r}} = \mathbf{a}_{\mathrm{other}} + \mathbf{a}_{gm}\). The \(\beta_i\) can be estimated as static parameters via ("beta_fogm", spice_id) state entries.

In deterministic propagation only the drift term is used:

\[\dot{\mathbf a}_{gm} = -\mathbf B \, \mathbf a_{gm}\]

The continuous-time process noise intensity consistent with the stationary first-order Gauss-Markov model is:

\[q_i = 2 \, \beta_i \, \sigma_i^2\]

where \(\sigma_i^2\) is the stationary variance of the corresponding acceleration component.

Parameters:

state_array (StateArray) – State definition containing the a_fogm Gauss-Markov acceleration state and, optionally, the beta_fogm inverse correlation coefficients.
spacecraft (Spacecraft) – Spacecraft object associated with the propagated body.
base_frame (Frame) – Reference frame of the propagator.
default_beta (np.ndarray, optional) – Default inverse correlation coefficients \([\beta_x, \beta_y, \beta_z]\) in 1/TU. Used when beta_fogm is not found in the state array. Defaults to isotropic unit decay [1, 1, 1].

Notes

The state vector must contain entries of the form:

('a_fogm', 3, 'estimated', 'dynamic', owner, value)

Optionally, if the inverse correlation times are themselves estimated:

('beta_fogm', 3, 'estimated', 'static', owner, value)

The model is linear in \(\mathbf a_{gm}\) for fixed \(\mathbf B\). If process noise is needed in a filter, this class provides the deterministic drift and Jacobians; the user should inject the corresponding process noise covariance separately.

See also

scarabaeus.StateArray: State vector used to initialize the model.
scarabaeus.Propagator: Propagator that integrates the augmented state.
scarabaeus.DynamicModel: Abstract base class for all force model components.
scarabaeus.PiecewiseFirstOrderGaussMarkov: Batch-segmented variant of this model.

References

Maybeck, P. S. (1979). Stochastic Models, Estimation, and Control. Brown, R. G., & Hwang, P. Y. C. (2012). Introduction to Random Signals and Applied Kalman Filtering.

Examples

import scarabaeus as scb

state = [
    ("position", 3, "estimated", "dynamic", sc, [7000, 0, 0]),
    ("velocity", 3, "estimated", "dynamic", sc, [0, 7.5, 0]),
    ("a_fogm", 3, "estimated", "dynamic", sc, [1e-9, 0.0, 0.0]),
    ("beta_fogm", 3, "estimated", "static", sc, [1/3600, 1/3600, 1/3600]),
]

state_vector = scb.StateArray(
    epoch=epochs[0],
    frame="J2000",
    origin=scb.CelestialBody("Earth"),
    state=state,
)

gm_model = FirstOrderGaussMarkov(
    state_array=state_vector,
    spacecraft=sc,
    base_frame="J2000",
)

Attributes:

default_beta: Default inverse correlation coefficients used when not specified in state.
origin: Celestial body serving as origin of the state vector.
origin_name: Name of the origin body.
ref_frame: Reference frame used by the model.

Methods

compute_acceleration(position, ...)

Returns the current Gauss-Markov acceleration applied to the body.

compute_acceleration(position: ndarray, current_state: dict, epoch: float, body: Body) → ndarray#

Returns the current Gauss-Markov acceleration applied to the body.

Parameters:

position (np.ndarray) – Spacecraft position vector, unused in this model but kept for API consistency.
current_state (dict) – Dictionary containing the propagated state.
epoch (float) – Current epoch, unused in this model but kept for API consistency.
body (Body) – Body for which the acceleration is evaluated.

Returns:

Current acceleration vector \(\mathbf a_{gm}\) in km/s^2.

Return type:

np.ndarray

property default_beta: ndarray#: Default inverse correlation coefficients used when not specified in state.

property origin: Body#: Celestial body serving as origin of the state vector.

property origin_name: str#: Name of the origin body.

property ref_frame: str#: Reference frame used by the model.

FirstOrderGaussMarkov#

`FirstOrderGaussMarkov`#