navlie.lib.preintegration.PreintegratedBodyVelocity¶

class navlie.lib.preintegration.PreintegratedBodyVelocity¶

Bases: ProcessModel

Process model that performs prediction of the state given an RMI \(\Delta \mathbf{U}_{ij}\) according to the equation

\[\mathbf{T}_{j} = \mathbf{T}_{i} \Delta \mathbf{U}_{ij}.\]

The covariance is also propagated accordingly

evaluate(x: MatrixLieGroupState, rmi: BodyVelocityIncrement, dt=None) → MatrixLieGroupState¶

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: MatrixLieGroupState, rmi: BodyVelocityIncrement, dt=None) → 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: MatrixLieGroupState, rmi: BodyVelocityIncrement, dt=None) → 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

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.preintegration.PreintegratedAngularVelocity

navlie.lib.preintegration.PreintegratedIMUKinematics