Crate linearkalman [−] [src]
This crate implements a standard linear Kalman filter and smoothing for
vectors of arbitrary dimension. Implementation method relies on rulinalg
library for linear algebra computations. Most inputs and outputs rely
therefore on (derived) constructs from
rulinalg
library, in particular
Vector<f64>
and
Matrix<f64>
structs.
Currently, implementation method assumes that Kalman filter is time-invariant and is based on the equations detailed below. Notations in the below equations correspond to annotations in the source code.
Measurement and state equations:
z_{t} = H_{t} x_{t} + v_{t}
wherev_{t} ~ N(0, R_{t})
x_{t} = F_{t} x_{t-1} + B_{t} u_{t} + w_{t}
wherew_{t} ~ N(0, Q_{t})
Kalman filter equations:
P_{t|t-1} = F_{t} P_{t-1|t-1} F'_{t} + Q_{t}
x_{t|t-1} = F_{t} x_{t-1|t-1} + B_{t} u_{t}
K_{t} = P_{t|t-1} H'_{t} * (H_{t} P_{t|t-1} H'_{t} + R_{t})^{-1}
P_{t|t} = (Id - K_{t} H_{t}) * P_{t|t-1}
x_{t|t} = x_{t|t-1} + K_{t} * (z_{t} - H_{t} x_{t|t-1})
Kalman smoothing equations:
J_{t} = P_{t|t} F'_{t} P_{t+1|t}^{-1}
x_{t|T} = x_{t|t} + J_{t} * (x_{t+1|T} - x_{t+1|t})
P_{t|T} = P_{t|t} + J_{t} * (P_{t+1|T} - P_{t+1|t}) * J'_{t}
Nomenclature:
(x_{t+1|t}, P_{t+1|t})
will be referred to as predicted state variables.(x_{t|t}, P_{t|t})
will be referred to as filtered state variables.(x_{t|T}, P_{t|T})
will be referred to as smoothed state variables.
For now, it is assumed here that B_{t}
matrix is null and that Q_{t}, R_{t}, H_{t}
and F_{t}
matrices are constant over time.
Besides real data, algorithm takes as inputs H
, R
, F
and Q
matrices
as well as initial guesses for state mean x0 = x_{1|0}
and covariance
matrix P0 = P_{1|0}
. Covariance matrix P0
indicates the uncertainty
related to the guess of x0
.
Structs
KalmanFilter |
Container object with values for matrices used in the Kalman filtering procedure as well as initial values for state variables. |
KalmanState |
Container with the value of state variable and its covariance. This struct is used throughout all parts of Kalman procedure and may refer to predicted, filtered and smoothed variables depending on the context. |
Functions
filter_step |
Returns a tuple containing updated and predicted estimates (in that order)
of the state variable and its covariance. This function might be useful for
cases where data is incoming and being updated in real-time so that Kalman
filtering is run incrementally. Note that given some initial values for |
predict_step |
Returns a predicted state variable mean and covariance. If prediction for
time |
update_step |
Returns an updated state variable mean and covariance given predicted and observed data. Typically, update step will be called after prediction step, data of which will be consequently used as input in updating. |