"""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, print_function,
unicode_literals)
import numpy as np
import scipy.linalg as linalg
from numpy import dot, zeros, eye, isscalar
from filterpy.common import setter, setter_1d, setter_scalar, dot3
class KalmanFilter(object):
""" Implements a Kalman filter. You are responsible for setting the
various state variables to reasonable values; the defaults will
not give you a functional filter.
You will have to set the following attributes after constructing this
object for the filter to perform properly. Please note that there are
various checks in place to ensure that you have made everything the
'correct' size. However, it is possible to provide incorrectly sized
arrays such that the linear algebra can not perform an operation.
It can also fail silently - you can end up with matrices of a size that
allows the linear algebra to work, but are the wrong shape for the problem
you are trying to solve.
**Attributes**
x : numpy.array(dim_x, 1)
state estimate vector
P : numpy.array(dim_x, dim_x)
covariance estimate matrix
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_x, dim_x)
You may read the following attributes.
**Readable Attributes**
y : numpy.array
Residual of the update step.
K : numpy.array(dim_x, dim_z)
Kalman gain of the update step
S : numpy.array
Systen uncertaintly projected to measurement space
"""
def __init__(self, dim_x, dim_z, dim_u=0):
""" Create a Kalman filter. 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.
"""
assert dim_x > 0
assert dim_z > 0
assert dim_u >= 0
self.dim_x = dim_x
self.dim_z = dim_z
self.dim_u = dim_u
self._x = zeros((dim_x,1))
self._P = eye(dim_x)
self._Q = eye(dim_x)
self._B = 0
self._F = 0
self.H = 0
self.R = eye(dim_z)
self._alpha_sq = 1.
self._K = 0
self._y = zeros((dim_z, 1))
self._S = 0
self._I = np.eye(dim_x)
def update(self, z, R=None, H=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.
R : np.array, scalar, or None
Optionally provide R to override the measurement noise for this
one call, otherwise self.R will be used.
"""
if z is None:
return
if R is None:
R = self.R
elif isscalar(R):
R = eye(self.dim_z) * R
if H is None:
H = self.H
P = self._P
x = self._x
self._y = z - dot(H, x)
S = dot3(H, P, H.T) + R
K = dot3(P, H.T, linalg.inv(S))
self._x = x + dot(K, self._y)
I_KH = self._I - dot(K, H)
self._P = dot3(I_KH, P, I_KH.T) + dot3(K, R, K.T)
self._S = S
self._K = K
def test_matrix_dimensions(self):
""" Performs a series of asserts to check that the size of everything
is what it should be. This can help you debug problems in your design.
This is only a test; you do not need to use it while filtering.
However, to use you will want to perform at least one predict() and
one update() before calling; some bad designs will cause the shapes
of x and P to change in a silent and bad way. For example, if you
pass in a badly dimensioned z into update that can cause x to be
misshapen."""
assert self._x.shape == (self.dim_x, 1), \
"Shape of x must be ({},{}), but is {}".format(
self.dim_x, 1, self._x.shape)
assert self._P.shape == (self.dim_x, self.dim_x), \
"Shape of P must be ({},{}), but is {}".format(
self.dim_x, self.dim_x, self._P.shape)
assert self._Q.shape == (self.dim_x, self.dim_x), \
"Shape of P must be ({},{}), but is {}".format(
self.dim_x, self.dim_x, self._P.shape)
def predict(self, u=0):
""" Predict next position.
**Parameters**
u : np.array
Optional control vector. If non-zero, it is multiplied by B
to create the control input into the system.
"""
self._x = dot(self._F, self.x) + dot(self._B, u)
self._P = self._alpha_sq * dot3(self._F, self._P, self._F.T) + self._Q
def batch_filter(self, zs, Rs=None, update_first=False):
""" Batch processes a sequences of measurements.
**Parameters**
zs : list-like
list of measurements at each time step `self.dt` Missing
measurements must be represented by 'None'.
Rs : list-like, optional
optional list of values to use for the measurement error
covariance; a value of None in any position will cause the filter
to use `self.R` for that time step.
update_first : bool, optional,
controls whether the order of operations is update followed by
predict, or predict followed by update. Default is predict->update.
**Returns**
means: np.array((n,dim_x,1))
array of the state for each time step after the update. Each entry
is an np.array. In other words `means[k,:]` is the state at step
`k`.
covariance: np.array((n,dim_x,dim_x))
array of the covariances for each time step after the update.
In other words `covariance[k,:,:]` is the covariance at step `k`.
means_predictions: np.array((n,dim_x,1))
array of the state for each time step after the predictions. Each
entry is an np.array. In other words `means[k,:]` is the state at
step `k`.
covariance_predictions: np.array((n,dim_x,dim_x))
array of the covariances for each time step after the prediction.
In other words `covariance[k,:,:]` is the covariance at step `k`.
"""
try:
z = zs[0]
except:
assert not isscalar(zs), 'zs must be list-like'
if self.dim_z == 1:
assert isscalar(z) or (z.ndim==1 and len(z) == 1), \
'zs must be a list of scalars or 1D, 1 element arrays'
else:
assert len(z) == self.dim_z, 'each element in zs must be a'
'1D array of length {}'.format(self.dim_z)
n = np.size(zs,0)
if Rs is None:
Rs = [None]*n
if self.x.ndim == 1:
means = zeros((n, self.dim_x))
means_p = zeros((n, self.dim_x))
else:
means = zeros((n, self.dim_x, 1))
means_p = zeros((n, self.dim_x, 1))
covariances = zeros((n, self.dim_x, self.dim_x))
covariances_p = zeros((n, self.dim_x, self.dim_x))
if update_first:
for i, (z, r) in enumerate(zip(zs, Rs)):
self.update(z, r)
means[i,:] = self._x
covariances[i,:,:] = self._P
self.predict()
means_p[i,:] = self._x
covariances_p[i,:,:] = self._P
else:
for i, (z, r) in enumerate(zip(zs, Rs)):
self.predict()
means_p[i,:] = self._x
covariances_p[i,:,:] = self._P
self.update(z, r)
means[i,:] = self._x
covariances[i,:,:] = self._P
return (means, covariances, means_p, covariances_p)
def rts_smoother(self, Xs, Ps, Qs=None):
""" Runs the Rauch-Tung-Striebal Kalman smoother on a set of
means and covariances computed by a Kalman filter. The usual input
would come from the output of `KalmanFilter.batch_filter()`.
**Parameters**
Xs : numpy.array
array of the means (state variable x) of the output of a Kalman
filter.
Ps : numpy.array
array of the covariances of the output of a kalman filter.
Q : list-like collection of numpy.array, optional
Process noise of the Kalman filter at each time step. Optional,
if not provided the filter's self.Q will be used
**Returns**
'x' : numpy.ndarray
smoothed means
'P' : numpy.ndarray
smoothed state covariances
'K' : numpy.ndarray
smoother gain at each step
**Example**::
zs = [t + random.randn()*4 for t in range (40)]
(mu, cov, _, _) = kalman.batch_filter(zs)
(x, P, K) = rts_smoother(mu, cov, fk.F, fk.Q)
"""
assert len(Xs) == len(Ps)
shape = Xs.shape
n = shape[0]
dim_x = shape[1]
F = self._F
if not Qs:
Qs = [self.Q] * n
K = zeros((n,dim_x,dim_x))
x, P = Xs.copy(), Ps.copy()
for k in range(n-2,-1,-1):
P_pred = dot3(F, P[k], F.T) + Qs[k]
K[k] = dot3(P[k], F.T, linalg.inv(P_pred))
x[k] += dot (K[k], x[k+1] - dot(F, x[k]))
P[k] += dot3 (K[k], P[k+1] - P_pred, K[k].T)
return (x, P, K)
def get_prediction(self, u=0):
""" Predicts the next state of the filter and returns it. Does not
alter the state of the filter.
**Parameters**
u : np.array
optional control input
**Returns**
(x, P)
State vector and covariance array of the prediction.
"""
x = dot(self._F, self._x) + dot(self._B, u)
P = self._alpha_sq * dot3(self._F, self._P, self._F.T) + self._Q
return (x, P)
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)
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 alpha(self):
""" Fading memory setting. 1.0 gives the normal Kalman filter, and
values slightly larger than 1.0 (such as 1.02) give a fading
memory effect - previous measurements have less influence on the
filter's estimates. This formulation of the Fading memory filter
(there are many) is due to Dan Simon [1].
** References **
[1] Dan Simon. "Optimal State Estimation." John Wiley & Sons.
p. 208-212. (2006)
"""
return self._alpha_sq**.5
@alpha.setter
def alpha(self, value):
assert np.isscalar(value)
assert value > 0
self._alpha_sq = value**2
@property
def Q(self):
""" Process uncertainty"""
return self._Q
@Q.setter
def Q(self, value):
self._Q = setter_scalar(value, self.dim_x)
@property
def P(self):
""" covariance matrix"""
return self._P
@P.setter
def P(self, value):
self._P = setter_scalar(value, self.dim_x)
@property
def F(self):
""" state transition matrix"""
return self._F
@F.setter
def F(self, value):
self._F = setter(value, self.dim_x, self.dim_x)
@property
def B(self):
""" control transition matrix"""
return self._B
@B.setter
def B(self, value):
""" control transition matrix"""
if np.isscalar(value):
self._B = value
else:
self._B = setter (value, self.dim_x, self.dim_u)
@property
def x(self):
""" filter state vector."""
return self._x
@x.setter
def x(self, value):
self._x = setter_1d(value, self.dim_x)
@property
def K(self):
""" Kalman gain """
return self._K
@property
def y(self):
""" measurement residual (innovation) """
return self._y
@property
def S(self):
""" system uncertainty in measurement space """
return self._S
