# SquareRootKalmanFilter¶

## Introduction and Overview¶

This implements a square root Kalman filter. No real attempt has been made to make this fast; it is a pedalogical exercise. The idea is that by computing and storing the square root of the covariance matrix we get about double the significant number of bits. Some authors consider this somewhat unnecessary with modern hardware. Of course, with microcontrollers being all the rage these days, that calculus has changed. But, will you really run a Kalman filter in Python on a tiny chip? No. So, this is for learning.

Copyright 2015 Roger R Labbe Jr.

FilterPy library. http://github.com/rlabbe/filterpy

Supporting book at: https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python

Square Root Kalman Filter

class filterpy.kalman.SquareRootKalmanFilter(dim_x, dim_z, dim_u=0)[source]

Create a Kalman filter which uses a square root implementation. This uses the square root of the state covariance matrix, which doubles the numerical precision of the filter, Therebuy reducing the effect of round off errors.

It is likely that you do not need to use this algorithm; we understand divergence issues very well now. However, if you expect the covariance matrix P to vary by 20 or more orders of magnitude then perhaps this will be useful to you, as the square root will vary by 10 orders of magnitude. From my point of view this is merely a ‘reference’ algorithm; I have not used this code in real world software. Brown has a useful discussion of when you might need to use the square root form of this algorithm.

You are responsible for setting the various state variables to reasonable values; the defaults below will not give you a functional filter.

Parameters: dim_x : int Number of state variables for the Kalman filter. For example, if you are tracking the position and velocity of an object in two dimensions, dim_x would be 4. This is used to set the default size of P, Q, and u dim_z : int Number of of measurement inputs. For example, if the sensor provides you with position in (x,y), dim_z would be 2. dim_u : int (optional) size of the control input, if it is being used. Default value of 0 indicates it is not used.

References

 Robert Grover Brown. Introduction to Random Signals and Applied
Kalman Filtering. Wiley and sons, 2012.

Examples

See my book Kalman and Bayesian Filters in Python https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python

Attributes: x : numpy.array(dim_x, 1) State estimate P : numpy.array(dim_x, dim_x) covariance matrix x_prior : numpy.array(dim_x, 1) Prior (predicted) state estimate. The *_prior and *_post attributes are for convienence; they store the prior and posterior of the current epoch. Read Only. P_prior : numpy.array(dim_x, dim_x) covariance matrix of the prior x_post : numpy.array(dim_x, 1) Posterior (updated) state estimate. Read Only. P_post : numpy.array(dim_x, dim_x) covariance matrix of the posterior z : numpy.array Last measurement used in update(). Read only. R : numpy.array(dim_z, dim_z) measurement uncertainty Q : numpy.array(dim_x, dim_x) Process uncertainty F : numpy.array() State Transition matrix H : numpy.array(dim_z, dim_x) Measurement function y : numpy.array Residual of the update step. Read only. K : numpy.array(dim_x, dim_z) Kalman gain of the update step. Read only. S : numpy.array Systen uncertaintly projected to measurement space. Read only.
__init__(dim_x, dim_z, dim_u=0)[source]

x.__init__(…) initializes x; see help(type(x)) for signature

update(z, R2=None)[source]

Add a new measurement (z) to the kalman filter. If z is None, nothing is changed.

Parameters: z : np.array measurement for this update. R2 : np.array, scalar, or None Sqrt of meaaurement noize. Optionally provide to override the measurement noise for this one call, otherwise self.R2 will be used.
predict(u=0)[source]

Predict next state (prior) using the Kalman filter state propagation equations.

Parameters: u : np.array, optional Optional control vector. If non-zero, it is multiplied by B to create the control input into the system.
residual_of(z)[source]

returns the residual for the given measurement (z). Does not alter the state of the filter.

measurement_of_state(x)[source]

Helper function that converts a state into a measurement.

Parameters: x : np.array kalman state vector z : np.array measurement corresponding to the given state
Q1_2

Sqrt Process uncertainty

Q

Process uncertainty

P_prior

covariance matrix of the prior

P_post

covariance matrix of the posterior

P1_2

sqrt of covariance matrix

P

covariance matrix

R1_2

sqrt of measurement uncertainty

R

measurement uncertainty