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FilterPy Source
For your convienence I have loaded several of FilterPy's core algorithms into this appendix.
KalmanFilter
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# %load https://raw.githubusercontent.com/rlabbe/filterpy/master/filterpy/kalman/kalman_filter.py """Copyright 2014 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)) # state self._P = eye(dim_x) # uncertainty covariance self._Q = eye(dim_x) # process uncertainty self._B = 0 # control transition matrix self._F = 0 # state transition matrix self.H = 0 # Measurement function self.R = eye(dim_z) # state uncertainty self._alpha_sq = 1. # fading memory control # gain and residual are computed during the innovation step. We # save them so that in case you want to inspect them for various # purposes self._K = 0 # kalman gain self._y = zeros((dim_z, 1)) self._S = 0 # system uncertainty in measurement space # identity matrix. Do not alter this. 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 # rename for readability and a tiny extra bit of speed if H is None: H = self.H P = self._P x = self._x # y = z - Hx # error (residual) between measurement and prediction self._y = z - dot(H, x) # S = HPH' + R # project system uncertainty into measurement space S = dot3(H, P, H.T) + R # K = PH'inv(S) # map system uncertainty into kalman gain K = dot3(P, H.T, linalg.inv(S)) # x = x + Ky # predict new x with residual scaled by the kalman gain self._x = x + dot(K, self._y) # P = (I-KH)P(I-KH)' + KRK' 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. """ # x = Fx + Bu self._x = dot(self._F, self.x) + dot(self._B, u) # P = FPF' + Q 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 # mean estimates from Kalman Filter 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)) # state covariances from Kalman Filter 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 # smoother gain 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""" 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
ExtendedKalmanFilter
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# %load https://raw.githubusercontent.com/rlabbe/filterpy/master/filterpy/kalman/EKF.py """Copyright 2014 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 from filterpy.common import setter, setter_1d, setter_scalar, dot3 class ExtendedKalmanFilter(object): def __init__(self, dim_x, dim_z, dim_u=0): """ Extended 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. """ self.dim_x = dim_x self.dim_z = dim_z self._x = zeros((dim_x,1)) # state self._P = eye(dim_x) # uncertainty covariance self._B = 0 # control transition matrix self._F = 0 # state transition matrix self._R = eye(dim_z) # state uncertainty self._Q = eye(dim_x) # process uncertainty self._y = zeros((dim_z, 1)) # identity matrix. Do not alter this. self._I = np.eye(dim_x) def predict_update(self, z, HJacobian, Hx, u=0): """ Performs the predict/update innovation of the extended Kalman filter. **Parameters** z : np.array measurement for this step. If `None`, only predict step is perfomed. HJacobian : function function which computes the Jacobian of the H matrix (measurement function). Takes state variable (self.x) as input, returns H. Hx : function function which takes a state variable and returns the measurement that would correspond to that state. u : np.array or scalar optional control vector input to the filter. """ if np.isscalar(z) and self.dim_z == 1: z = np.asarray([z], float) F = self._F B = self._B P = self._P Q = self._Q R = self._R x = self._x H = HJacobian(x) # predict step x = dot(F, x) + dot(B, u) P = dot3(F, P, F.T) + Q # update step S = dot3(H, P, H.T) + R K = dot3(P, H.T, linalg.inv (S)) self._x = x + dot(K, (z - Hx(x))) I_KH = self._I - dot(K, H) self._P = dot3(I_KH, P, I_KH.T) + dot3(K, R, K.T) def update(self, z, HJacobian, Hx, R=None): """ Performs the update innovation of the extended Kalman filter. **Parameters** z : np.array measurement for this step. If `None`, only predict step is perfomed. HJacobian : function function which computes the Jacobian of the H matrix (measurement function). Takes state variable (self.x) as input, returns H. Hx : function function which takes a state variable and returns the measurement that would correspond to that state. """ P = self._P if R is None: R = self._R elif np.isscalar(R): R = eye(self.dim_z) * R if np.isscalar(z) and self.dim_z == 1: z = np.asarray([z], float) x = self._x H = HJacobian(x) S = dot3(H, P, H.T) + R K = dot3(P, H.T, linalg.inv (S)) y = z - Hx(x) self._x = x + dot(K, y) I_KH = self._I - dot(K, H) self._P = dot3(I_KH, P, I_KH.T) + dot3(K, R, K.T) def predict_x(self, u=0): """ predicts the next state of X. If you need to compute the next state yourself, override this function. You would need to do this, for example, if the usual Taylor expansion to generate F is not providing accurate results for you. """ self._x = dot(self._F, self._x) + dot(self._B, u) 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.predict_x(u) self._P = dot3(self._F, self._P, self._F.T) + self._Q @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 R(self): """ measurement uncertainty""" return self._R @R.setter def R(self, value): self._R = setter_scalar(value, self.dim_z) @property def F(self): return self._F @F.setter def F(self, value): self._F = setter(value, self.dim_x, self.dim_x) @property def B(self): return self._B @B.setter def B(self, value): """ control transition matrix""" self._B = setter(value, self.dim_x, self.dim_u) @property def x(self): 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
UnscentedKalmanFilter
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# %load https://raw.githubusercontent.com/rlabbe/filterpy/master/filterpy/kalman/UKF.py """Copyright 2014 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. """ # pylint bug - warns about numpy functions which do in fact exist. # pylint: disable=E1101 #I like aligning equal signs for readability of math # pylint: disable=C0326 from __future__ import (absolute_import, division, print_function, unicode_literals) from numpy.linalg import inv, cholesky import numpy as np from numpy import asarray, eye, zeros, dot, isscalar, outer from filterpy.common import dot3 class UnscentedKalmanFilter(object): # pylint: disable=too-many-instance-attributes # pylint: disable=C0103 """ Implements the Unscented Kalman filter (UKF) as defined by Simon J. Julier and Jeffery K. Uhlmann [1]. Succintly, the UKF selects a set of sigma points and weights inside the covariance matrix of the filter's state. These points are transformed through the nonlinear process being filtered, and are rebuilt into a mean and covariance by computed the weighted mean and expected value of the transformed points. Read the paper; it is excellent. My book "Kalman and Bayesian Filters in Python" [2] explains the algorithm, develops this code, and provides examples of the filter in use. You will have to set the following attributes after constructing this object for the filter to perform properly. **Attributes** x : numpy.array(dim_x) 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 You may read the following attributes. **Readable Attributes** Pxz : numpy.aray(dim_x, dim_z) Cross variance of x and z computed during update() call. **References** .. [1] Julier, Simon J.; Uhlmann, Jeffrey "A New Extension of the Kalman Filter to Nonlinear Systems". Proc. SPIE 3068, Signal Processing, Sensor Fusion, and Target Recognition VI, 182 (July 28, 1997) .. [2] Labbe, Roger R. "Kalman and Bayesian Filters in Python" https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python """ def __init__(self, dim_x, dim_z, dt, hx, fx, kappa=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 filter. For example, if you are tracking the position and velocity of an object in two dimensions, dim_x would be 4. 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. dt : float Time between steps in seconds. hx : function(x) Measurement function. Converts state vector x into a measurement vector of shape (dim_z). fx : function(x,dt) function that returns the state x transformed by the state transistion function. dt is the time step in seconds. kappa : float, default=0. Scaling factor that can reduce high order errors. kappa=0 gives the standard unscented filter. According to [1], if you set kappa to 3-dim_x for a Gaussian x you will minimize the fourth order errors in x and P. **References** [1] S. Julier, J. Uhlmann, and H. Durrant-Whyte. "A new method for the nonlinear transformation of means and covariances in filters and estimators," IEEE Transactions on Automatic Control, 45(3), pp. 477-482 (March 2000). """ self.Q = eye(dim_x) self.R = eye(dim_z) self.x = zeros(dim_x) self.P = eye(dim_x) self._dim_x = dim_x self._dim_z = dim_z self._dt = dt self._num_sigmas = 2*dim_x + 1 self.kappa = kappa self.hx = hx self.fx = fx # weights for the sigma points self.W = self.weights(dim_x, kappa) # sigma points transformed through f(x) and h(x) # variables for efficiency so we don't recreate every update self.sigmas_f = zeros((self._num_sigmas, self._dim_x)) def update(self, z, R=None, residual=np.subtract, UT=None): """ Update the UKF with the given measurements. On return, self.x and self.P contain the new mean and covariance of the filter. **Parameters** z : numpy.array of shape (dim_z) measurement vector R : numpy.array((dim_z, dim_z)), optional Measurement noise. If provided, overrides self.R for this function call. residual : function (z, z2), optional Optional function that computes the residual (difference) between the two measurement vectors. If you do not provide this, then the built in minus operator will be used. You will normally want to use the built in unless your residual computation is nonlinear (for example, if they are angles) UT : function(sigmas, Wm, Wc, noise_cov), optional Optional function to compute the unscented transform for the sigma points passed through hx. Typically the default function will work, but if for example you are using angles the default method of computing means and residuals will not work, and you will have to define how to compute it. """ if isscalar(z): dim_z = 1 else: dim_z = len(z) if R is None: R = self.R elif np.isscalar(R): R = eye(self._dim_z) * R # rename for readability sigmas_f = self.sigmas_f sigmas_h = zeros((self._num_sigmas, dim_z)) if UT is None: UT = unscented_transform # transform sigma points into measurement space for i in range(self._num_sigmas): sigmas_h[i] = self.hx(sigmas_f[i]) # mean and covariance of prediction passed through inscented transform zp, Pz = UT(sigmas_h, self.W, self.W, R) # compute cross variance of the state and the measurements '''self.Pxz = zeros((self._dim_x, dim_z)) for i in range(self._num_sigmas): self.Pxz += self.W[i] * np.outer(sigmas_f[i] - self.x, residual(sigmas_h[i], zp))''' # this is the unreadable but fast implementation of the # commented out loop above yh = sigmas_f - self.x[np.newaxis, :] yz = residual(sigmas_h, zp[np.newaxis, :]) self.Pxz = yh.T.dot(np.diag(self.W)).dot(yz) K = dot(self.Pxz, inv(Pz)) # Kalman gain y = residual(z, zp) self.x = self.x + dot(K, y) self.P = self.P - dot3(K, Pz, K.T) def predict(self, dt=None): """ Performs the predict step of the UKF. On return, self.xp and self.Pp contain the predicted state (xp) and covariance (Pp). 'p' stands for prediction. **Parameters** dt : double, optional If specified, the time step to be used for this prediction. self._dt is used if this is not provided. Important: this MUST be called before update() is called for the first time. """ if dt is None: dt = self._dt # calculate sigma points for given mean and covariance sigmas = self.sigma_points(self.x, self.P, self.kappa) for i in range(self._num_sigmas): self.sigmas_f[i] = self.fx(sigmas[i], dt) self.x, self.P = unscented_transform( self.sigmas_f, self.W, self.W, self.Q) def batch_filter(self, zs, Rs=None, residual=np.subtract, UT=None): """ Performs the UKF filter over the list of measurement in `zs`. **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. residual : function (z, z2), optional Optional function that computes the residual (difference) between the two measurement vectors. If you do not provide this, then the built in minus operator will be used. You will normally want to use the built in unless your residual computation is nonlinear (for example, if they are angles) UT : function(sigmas, Wm, Wc, noise_cov), optional Optional function to compute the unscented transform for the sigma points passed through hx. Typically the default function will work, but if for example you are using angles the default method of computing means and residuals will not work, and you will have to define how to compute it. **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`. """ 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 # mean estimates from Kalman Filter if self.x.ndim == 1: means = zeros((n, self._dim_x)) else: means = zeros((n, self._dim_x, 1)) # state covariances from Kalman Filter covariances = zeros((n, self._dim_x, self._dim_x)) for i, (z, r) in enumerate(zip(zs, Rs)): self.predict() self.update(z, r) means[i,:] = self.x covariances[i,:,:] = self.P return (means, covariances) def rts_smoother(self, Xs, Ps, Qs=None, dt=None): """ Runs the Rauch-Tung-Striebal Kalman smoother on a set of means and covariances computed by the UKF. The usual input would come from the output of `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 dt : optional, float or array-like of float If provided, specifies the time step of each step of the filter. If float, then the same time step is used for all steps. If an array, then each element k contains the time at step k. Units are seconds. **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) n, dim_x = Xs.shape if dt is None: dt = [self._dt] * n elif isscalar(dt): dt = [dt] * n if Qs is None: Qs = [self.Q] * n # smoother gain Ks = zeros((n,dim_x,dim_x)) num_sigmas = 2*dim_x + 1 xs, ps = Xs.copy(), Ps.copy() sigmas_f = zeros((num_sigmas, dim_x)) for k in range(n-2,-1,-1): # create sigma points from state estimate, pass through state func sigmas = self.sigma_points(xs[k], ps[k], self.kappa) for i in range(num_sigmas): sigmas_f[i] = self.fx(sigmas[i], dt[k]) # compute backwards prior state and covariance xb = dot(self.W, sigmas_f) Pb = 0 x = Xs[k] for i in range(num_sigmas): y = sigmas_f[i] - x Pb += self.W[i] * outer(y, y) Pb += Qs[k] # compute cross variance Pxb = 0 for i in range(num_sigmas): z = sigmas[i] - Xs[k] y = sigmas_f[i] - xb Pxb += self.W[i] * outer(z, y) # compute gain K = dot(Pxb, inv(Pb)) # update the smoothed estimates xs[k] += dot (K, xs[k+1] - xb) ps[k] += dot3(K, ps[k+1] - Pb, K.T) Ks[k] = K return (xs, ps, Ks) @staticmethod def weights(n, kappa): """ Computes the weights for an unscented Kalman filter. See __init__() for meaning of parameters. """ assert n > 0, "n must be greater than 0, it's value is {}".format(n) k = .5 / (n+kappa) W = np.full(2*n+1, k) W[0] = kappa / (n+kappa) return W @staticmethod def sigma_points(x, P, kappa): """ Computes the sigma points for an unscented Kalman filter given the mean (x) and covariance(P) of the filter. kappa is an arbitrary constant. Returns sigma points. Works with both scalar and array inputs: sigma_points (5, 9, 2) # mean 5, covariance 9 sigma_points ([5, 2], 9*eye(2), 2) # means 5 and 2, covariance 9I **Parameters** X An array-like object of the means of length n Can be a scalar if 1D. examples: 1, [1,2], np.array([1,2]) P : scalar, or np.array Covariance of the filter. If scalar, is treated as eye(n)*P. kappa : float Scaling factor. **Returns** sigmas : np.array, of size (n, 2n+1) 2D array of sigma points. Each column contains all of the sigmas for one dimension in the problem space. They are ordered as: .. math:: sigmas[0] = x \n sigmas[1..n] = x + [\sqrt{(n+\kappa)P}]_k \n sigmas[n+1..2n] = x - [\sqrt{(n+\kappa)P}]_k """ if np.isscalar(x): x = asarray([x]) n = np.size(x) # dimension of problem if np.isscalar(P): P = eye(n)*P sigmas = zeros((2*n+1, n)) # implements U'*U = (n+kappa)*P. Returns lower triangular matrix. # Take transpose so we can access with U[i] U = cholesky((n+kappa)*P).T #U = sqrtm((n+kappa)*P).T sigmas[0] = x sigmas[1:n+1] = x + U sigmas[n+1:2*n+2] = x - U return sigmas def unscented_transform(Sigmas, Wm, Wc, noise_cov): """ Computes unscented transform of a set of sigma points and weights. returns the mean and covariance in a tuple. """ kmax, n = Sigmas.shape # new mean is just the sum of the sigmas * weight x = dot(Wm, Sigmas) # dot = \Sigma^n_1 (W[k]*Xi[k]) # new covariance is the sum of the outer product of the residuals # times the weights '''P = zeros((n, n)) for k in range(kmax): y = Sigmas[k] - x P += Wc[k] * np.outer(y, y)''' # this is the fast way to do the commented out code above y = Sigmas - x[np.newaxis,:] P = y.T.dot(np.diag(Wc)).dot(y) if noise_cov is not None: P += noise_cov return (x, P)