# -*- coding: utf-8 -*-
# pylint: disable=invalid-name, too-many-instance-attributes
"""Copyright 2015 Roger R Labbe Jr.
FilterPy library.
http://github.com/rlabbe/filterpy
Documentation at:
https://filterpy.readthedocs.org
Supporting book at:
https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python
This is licensed under an MIT license. See the readme.MD file
for more information.
"""
from __future__ import (absolute_import, division)
from copy import deepcopy
import numpy as np
from numpy import dot, zeros, eye
from scipy.linalg import cholesky, qr, pinv
from filterpy.common import pretty_str
[docs]class SquareRootKalmanFilter(object):
"""
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[1]
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.
Attributes
----------
x : numpy.array(dim_x, 1)
State estimate
P : numpy.array(dim_x, dim_x)
State 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)
Prior (predicted) state covariance matrix. Read Only.
x_post : numpy.array(dim_x, 1)
Posterior (updated) state estimate. Read Only.
P_post : numpy.array(dim_x, dim_x)
Posterior (updated) state covariance matrix. Read Only.
z : numpy.array
Last measurement used in update(). Read only.
R : numpy.array(dim_z, dim_z)
Measurement noise matrix
Q : numpy.array(dim_x, dim_x)
Process noise matrix
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.
Examples
--------
See my book Kalman and Bayesian Filters in Python
https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python
References
----------
[1] Robert Grover Brown. Introduction to Random Signals and Applied
Kalman Filtering. Wiley and sons, 2012.
"""
[docs] def __init__(self, dim_x, dim_z, dim_u=0):
if dim_z < 1:
raise ValueError('dim_x must be 1 or greater')
if dim_z < 1:
raise ValueError('dim_x must be 1 or greater')
if dim_u < 0:
raise ValueError('dim_x must be 0 or greater')
self.dim_x = dim_x
self.dim_z = dim_z
self.dim_u = dim_u
self.x = zeros((dim_x, 1)) # state
self._P = eye(dim_x) # uncertainty covariance
self._P1_2 = eye(dim_x) # sqrt uncertainty covariance
self._Q = eye(dim_x) # sqrt process uncertainty
self._Q1_2 = eye(dim_x) # sqrt process uncertainty
self.B = 0. # control transition matrix
self.F = np.eye(dim_x) # state transition matrix
self.H = np.zeros((dim_z, dim_x)) # Measurement function
self._R1_2 = eye(dim_z) # sqrt state uncertainty
self._R = eye(dim_z) # state uncertainty
self.z = np.array([[None]*self.dim_z]).T
self.K = 0.
self.S = 0.
# Residual is computed during the innovation (update) step. We
# save it so that in case you want to inspect it for various
# purposes
self.y = zeros((dim_z, 1))
# identity matrix.
self._I = np.eye(dim_x)
self.M = np.zeros((dim_z + dim_x, dim_z + dim_x))
# copy prior and posterior
self.x_prior = np.copy(self.x)
self._P1_2_prior = np.copy(self._P1_2)
self.x_post = np.copy(self.x)
self._P1_2_post = np.copy(self._P1_2)
[docs] def update(self, z, R2=None):
"""
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.
"""
if z is None:
self.z = np.array([[None]*self.dim_z]).T
self.x_post = self.x.copy()
self._P1_2_post = np.copy(self._P1_2)
return
if R2 is None:
R2 = self._R1_2
elif np.isscalar(R2):
R2 = eye(self.dim_z) * R2
# rename for convienance
dim_z = self.dim_z
M = self.M
M[0:dim_z, 0:dim_z] = R2.T
M[dim_z:, 0:dim_z] = dot(self.H, self._P1_2).T
M[dim_z:, dim_z:] = self._P1_2.T
_, self.S = qr(M)
self.K = self.S[0:dim_z, dim_z:].T
N = self.S[0:dim_z, 0:dim_z].T
# y = z - Hx
# error (residual) between measurement and prediction
self.y = z - dot(self.H, self.x)
# x = x + Ky
# predict new x with residual scaled by the kalman gain
self.x += dot(self.K, pinv(N)).dot(self.y)
self._P1_2 = self.S[dim_z:, dim_z:].T
self.z = deepcopy(z)
self.x_post = self.x.copy()
self._P1_2_post = np.copy(self._P1_2)
[docs] def predict(self, u=0):
"""
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.
"""
# x = Fx + Bu
self.x = dot(self.F, self.x) + dot(self.B, u)
# P = FPF' + Q
_, P2 = qr(np.hstack([dot(self.F, self._P1_2), self._Q1_2]).T)
self._P1_2 = P2[:self.dim_x, :self.dim_x].T
# copy prior
self.x_prior = np.copy(self.x)
self._P1_2_prior = np.copy(self._P1_2)
[docs] def residual_of(self, z):
""" returns the residual for the given measurement (z). Does not alter
the state of the filter.
"""
return z - dot(self.H, self.x)
[docs] def measurement_of_state(self, x):
""" Helper function that converts a state into a measurement.
Parameters
----------
x : np.array
kalman state vector
Returns
-------
z : np.array
measurement corresponding to the given state
"""
return dot(self.H, x)
@property
def Q(self):
""" Process uncertainty"""
return dot(self._Q1_2.T, self._Q1_2)
@property
def Q1_2(self):
""" Sqrt Process uncertainty"""
return self._Q1_2
@Q.setter
def Q(self, value):
""" Process uncertainty"""
self._Q = value
self._Q1_2 = cholesky(self._Q, lower=True)
@property
def P(self):
""" covariance matrix"""
return dot(self._P1_2.T, self._P1_2)
@property
def P_prior(self):
""" covariance matrix of the prior"""
return dot(self._P1_2_prior.T, self._P1_2_prior)
@property
def P_post(self):
""" covariance matrix of the posterior"""
return dot(self._P1_2_prior.T, self._P1_2_prior)
@property
def P1_2(self):
""" sqrt of covariance matrix"""
return self._P1_2
@P.setter
def P(self, value):
""" covariance matrix"""
self._P = value
self._P1_2 = cholesky(self._P, lower=True)
@property
def R(self):
""" measurement uncertainty"""
return dot(self._R1_2.T, self._R1_2)
@property
def R1_2(self):
""" sqrt of measurement uncertainty"""
return self._R1_2
@R.setter
def R(self, value):
""" measurement uncertainty"""
self._R = value
self._R1_2 = cholesky(self._R, lower=True)
def __repr__(self):
return '\n'.join([
'SquareRootKalmanFilter object',
pretty_str('dim_x', self.dim_x),
pretty_str('dim_z', self.dim_z),
pretty_str('dim_u', self.dim_u),
pretty_str('x', self.x),
pretty_str('P', self.P),
pretty_str('F', self.F),
pretty_str('Q', self.Q),
pretty_str('R', self.R),
pretty_str('H', self.H),
pretty_str('K', self.K),
pretty_str('y', self.y),
pretty_str('S', self.S),
pretty_str('M', self.M),
pretty_str('B', self.B),
])