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# Kalman Filter

Fundamentals
Title Kalman Filter
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Year of Publication 2011

The principle of Kalman filtering can be roughly summarised as the weighted least square solution of the linearised observation system augmented with a prediction of the estimate as additional equations. The predicted estimate and the weighted solution are given as follows:

• Predicted estimate (from a simple linear model):
Let  be the estimate for the -th epoch, thence, a prediction for the next epoch , is computed according to the model [footnotes 1]:


Where  is named the Transition Matrix and defines the propagation of the vector parameters estimate , and  is the Process Noise Matrix. Matrix  allows, in particular, to add some additional noise to account for possible miss-modelling due to the simple prediction model used or, what is the same, to an inexact description of the problem in general.

• Weighted solution (from measurements and predicted estimate):
According to the approach of Block-Wise Weighted Least Square the measurements (i.e., the linearised observation equations) are combined with the predicted parameters estimate as follows:


which is solved as (see Block-Wise Weighted Least Square)


being the weighted least squares estimate:


The algorithm can be summarised in the following scheme [footnotes 2]:
Figure 1: Kalman filter diagram. Notation: RR, PP.

Using the following relations [1],


it can be shown that the previous formulation is algebraically equivalent to the classical formulation of the Kalman filter given by the following figure 2:
Figure 2: Classical formulation of Kalman filter

## Some elemental examples of matrix definitions  and 

The determination of the state transition matrix  and Process Noise matrix  is usually based on physical models describing the estimation problem. For instance, for satellite tracking or orbit determination, they are derived from the orbital motion equations. Elemental formulations, i.e., for the SPS and PPP navigation, are covered by the examples given as follows:

### Static positioning

The state vector to determine is given by  where the coordinates [footnotes 3] are considered as constants (because the receiver is kept fixed) and the clock offset can be modelled as a white noise with zero mean. Under these conditions matrix  and  are given by:



being  the uncertainty in the clock prediction model (for instance  for a unknown clock --i.e.  light-millisecond). Notice that the prediction model for the coordinates is exact and, thence, the associated elements in matrix  are null.

### Kinematic positioning

• If it is a vehicle running at a high velocity, the coordinates can be modelled as a white noise with zero mean, the same as the clock offset:


• If it is a vehicle running at a low velocity, the coordinates can be modelled as a {\it random walk} process with its uncertainty growing with time:


## Notes

1. ^ It is a first order model of Gauss-Markov. The dynamical character is established through the state transition matrix  and the noise matrix of the process .
2. ^ If one desires to go deeply into the theme, the book [Bierman, 1976] is recommended. Special chapters relating to U-D covariance filter and SRIF.
3. ^ We are referring to deviations from nominal values , that is what it is estimated from the navigation equations.

## References

1. ^ [Bierman, 1976] Bierman, G., 1976. Factorization Methods for Discrete Sequential estimation. Academic Press, New York, New York, USA.