navlie.lib.models.DoubleIntegratorWithBias¶

class navlie.lib.models.DoubleIntegratorWithBias(Q: ndarray)¶

Bases: DoubleIntegrator

The double-integrator process model, but with an additional bias on the input.

\[ \begin{align}\begin{aligned}\dot{\mathbf{r}} = \mathbf{v}\\\dot{\mathbf{v}} = \mathbf{u} - \mathbf{b}\\\dot{\mathbf{b}} = \mathbf{w}\end{aligned}\end{align} \]

where \(\mathbf{u}\) is the input and \(\mathbf{b}\) is the bias. The bias is assumed to be a random walk with covariance \(\mathbf{Q}\).

Parameters:: Q (np.ndarray) – Q: Discrete time covariance on the input u.
Raises:: ValueError – if Q is not an n x n matrix.

evaluate(x: VectorState, u: VectorInput, dt: float) → VectorState¶

Implementation of \({f}(\mathcal{X}_{k-1}, \mathbf{u}, \Delta t)\).

Parameters:

x (State) – State at time \(k-1\).
u (Input) – The input value \(\mathbf{u}\) provided as a Input object. The actual numerical value is accessed via u.value.
dt (float) – The time interval \(\Delta t\) between the two states.

Returns:

State at time \(k\).

Return type:

State

jacobian(x, u, dt) → ndarray¶

Implementation of the process model Jacobian with respect to the state.

\[\mathbf{F} = \frac{D {f}(\mathcal{X}_{k-1}, \mathbf{u}, \Delta t)}{D \mathcal{X}_{k-1}}\]

Parameters:

x (State) – State at time \(k-1\).
u (Input) – The input value \(\mathbf{u}\) provided as a Input object.
dt (float) – The time interval \(\Delta t\) between the two states.

Returns:

Process model Jacobian with respect to the state \(\mathbf{F}\).

Return type:

np.ndarray

covariance(x, u, dt) → ndarray¶

Covariance matrix \(\mathbf{Q}_k\) of the additive Gaussian noise \(\mathbf{w}_{k} \sim \mathcal{N}(\mathbf{0}, \mathbf{Q}_k)\). If this method is not overridden, the covariance of the process model error is approximated from the input covariance using a linearization procedure, with the input Jacobian evaluated using finite difference.

Parameters:

x (State) – State at time \(k-1\).
u (Input) – The input value \(\mathbf{u}\) provided as a Input object.
dt (float) – The time interval \(\Delta t\) between the two states.

Returns:

Covariance matrix \(\mathbf{Q}_k\).

Return type:

np.ndarray

input_jacobian(dt)¶

evaluate_with_jacobian(x: ~navlie.types.State, u: ~navlie.types.Input, dt: float) -> (<class 'navlie.types.State'>, <class 'numpy.ndarray'>)¶: Evaluates the process model and simultaneously returns the Jacobian as its second output argument. This is useful to override for performance reasons when the model evaluation and Jacobian have a lot of common calculations, and it is more efficient to calculate them in the same function call.

input_covariance(x: State, u: Input, dt: float) → ndarray¶

Covariance matrix of additive noise on the input.

Parameters:

x (State) – State at time \(k-1\).
u (Input) – The input value \(\mathbf{u}\) provided as a Input object.
dt (float) – The time interval \(\Delta t\) between the two states.

Returns:

Covariance matrix \(\mathbf{R}_k\).

Return type:

np.ndarray

input_jacobian_fd(x: State, u: Input, dt: float, step_size=1e-06, *args, **kwargs) → ndarray¶: Calculates the input jacobian with finite difference.

jacobian_fd(x: State, u: Input, dt: float, step_size=1e-06, *args, **kwargs) → ndarray¶: Calculates the model jacobian with finite difference.

sqrt_information(x: State, u: Input, dt: float) → ndarray¶

navlie.lib.models.DoubleIntegrator

navlie.lib.models.GlobalPosition