# -*- coding: utf-8 -*- # pylint: disable=invalid-name """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 from math import log, exp, sqrt import sys import numpy as np from numpy import eye, zeros, dot, isscalar, outer from scipy.linalg import cholesky from filterpy.kalman import unscented_transform from filterpy.stats import logpdf from filterpy.common import pretty_str class UnscentedKalmanFilter(object): # pylint: disable=too-many-instance-attributes # pylint: disable=invalid-name r""" Implements the Scaled Unscented Kalman filter (UKF) as defined by Simon Julier in [1], using the formulation provided by Wan and Merle in [2]. This filter scales the sigma points to avoid strong nonlinearities. 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. This is for convience, so everything is sized correctly on creation. If you are using multiple sensors the size of `z` can change based on the sensor. Just provide the appropriate hx function 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. points : class Class which computes the sigma points and weights for a UKF algorithm. You can vary the UKF implementation by changing this class. For example, MerweScaledSigmaPoints implements the alpha, beta, kappa parameterization of Van der Merwe, and JulierSigmaPoints implements Julier's original kappa parameterization. See either of those for the required signature of this class if you want to implement your own. sqrt_fn : callable(ndarray), default=None (implies scipy.linalg.cholesky) Defines how we compute the square root of a matrix, which has no unique answer. Cholesky is the default choice due to its speed. Typically your alternative choice will be scipy.linalg.sqrtm. Different choices affect how the sigma points are arranged relative to the eigenvectors of the covariance matrix. Usually this will not matter to you; if so the default cholesky() yields maximal performance. As of van der Merwe's dissertation of 2004 [6] this was not a well reseached area so I have no advice to give you. If your method returns a triangular matrix it must be upper triangular. Do not use numpy.linalg.cholesky - for historical reasons it returns a lower triangular matrix. The SciPy version does the right thing as far as this class is concerned. x_mean_fn : callable (sigma_points, weights), optional Function that computes the mean of the provided sigma points and weights. Use this if your state variable contains nonlinear values such as angles which cannot be summed. .. code-block:: Python def state_mean(sigmas, Wm): x = np.zeros(3) sum_sin, sum_cos = 0., 0. for i in range(len(sigmas)): s = sigmas[i] x[0] += s[0] * Wm[i] x[1] += s[1] * Wm[i] sum_sin += sin(s[2])*Wm[i] sum_cos += cos(s[2])*Wm[i] x[2] = atan2(sum_sin, sum_cos) return x z_mean_fn : callable (sigma_points, weights), optional Same as x_mean_fn, except it is called for sigma points which form the measurements after being passed through hx(). residual_x : callable (x, y), optional residual_z : callable (x, y), optional Function that computes the residual (difference) between x and y. You will have to supply this if your state variable cannot support subtraction, such as angles (359-1 degreees is 2, not 358). x and y are state vectors, not scalars. One is for the state variable, the other is for the measurement state. .. code-block:: Python def residual(a, b): y = a[0] - b[0] if y > np.pi: y -= 2*np.pi if y < -np.pi: y = 2*np.pi return y Attributes ---------- x : numpy.array(dim_x) state estimate vector P : numpy.array(dim_x, dim_x) covariance estimate matrix x_prior : numpy.array(dim_x) 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) Posterior (updated) state estimate. Read Only. P_post : numpy.array(dim_x, dim_x) Posterior (updated) state covariance matrix. Read Only. z : ndarray 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 K : numpy.array Kalman gain y : numpy.array innovation residual log_likelihood : scalar Log likelihood of last measurement update. likelihood : float likelihood of last measurment. Read only. Computed from the log-likelihood. The log-likelihood can be very small, meaning a large negative value such as -28000. Taking the exp() of that results in 0.0, which can break typical algorithms which multiply by this value, so by default we always return a number >= sys.float_info.min. mahalanobis : float mahalanobis distance of the measurement. Read only. inv : function, default numpy.linalg.inv If you prefer another inverse function, such as the Moore-Penrose pseudo inverse, set it to that instead: .. code-block:: Python kf.inv = np.linalg.pinv Examples -------- Simple example of a linear order 1 kinematic filter in 2D. There is no need to use a UKF for this example, but it is easy to read. >>> def fx(x, dt): >>> # state transition function - predict next state based >>> # on constant velocity model x = vt + x_0 >>> F = np.array([[1, dt, 0, 0], >>> [0, 1, 0, 0], >>> [0, 0, 1, dt], >>> [0, 0, 0, 1]], dtype=float) >>> return np.dot(F, x) >>> >>> def hx(x): >>> # measurement function - convert state into a measurement >>> # where measurements are [x_pos, y_pos] >>> return np.array([x[0], x[2]]) >>> >>> dt = 0.1 >>> # create sigma points to use in the filter. This is standard for Gaussian processes >>> points = MerweScaledSigmaPoints(4, alpha=.1, beta=2., kappa=-1) >>> >>> kf = UnscentedKalmanFilter(dim_x=4, dim_z=2, dt=dt, fx=fx, hx=hx, points=points) >>> kf.x = np.array([-1., 1., -1., 1]) # initial state >>> kf.P *= 0.2 # initial uncertainty >>> z_std = 0.1 >>> kf.R = np.diag([z_std**2, z_std**2]) # 1 standard >>> kf.Q = Q_discrete_white_noise(dim=2, dt=dt, var=0.01**2, block_size=2) >>> >>> zs = [[i+randn()*z_std, i+randn()*z_std] for i in range(50)] # measurements >>> for z in zs: >>> kf.predict() >>> kf.update(z) >>> print(kf.x, 'log-likelihood', kf.log_likelihood) For in depth explanations see my book Kalman and Bayesian Filters in Python https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python Also see the filterpy/kalman/tests subdirectory for test code that may be illuminating. References ---------- .. [1] Julier, Simon J. "The scaled unscented transformation," American Control Converence, 2002, pp 4555-4559, vol 6. Online copy: https://www.cs.unc.edu/~welch/kalman/media/pdf/ACC02-IEEE1357.PDF .. [2] E. A. Wan and R. Van der Merwe, “The unscented Kalman filter for nonlinear estimation,” in Proc. Symp. Adaptive Syst. Signal Process., Commun. Contr., Lake Louise, AB, Canada, Oct. 2000. Online Copy: https://www.seas.harvard.edu/courses/cs281/papers/unscented.pdf .. [3] 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). .. [4] E. A. Wan and R. Van der Merwe, “The Unscented Kalman filter for Nonlinear Estimation,” in Proc. Symp. Adaptive Syst. Signal Process., Commun. Contr., Lake Louise, AB, Canada, Oct. 2000. https://www.seas.harvard.edu/courses/cs281/papers/unscented.pdf .. [5] Wan, Merle "The Unscented Kalman Filter," chapter in *Kalman Filtering and Neural Networks*, John Wiley & Sons, Inc., 2001. .. [6] R. Van der Merwe "Sigma-Point Kalman Filters for Probabilitic Inference in Dynamic State-Space Models" (Doctoral dissertation) """ def __init__(self, dim_x, dim_z, dt, hx, fx, points, sqrt_fn=None, x_mean_fn=None, z_mean_fn=None, residual_x=None, residual_z=None): """ 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. """ #pylint: disable=too-many-arguments self.x = zeros(dim_x) self.P = eye(dim_x) self.x_prior = np.copy(self.x) self.P_prior = np.copy(self.P) self.Q = eye(dim_x) self.R = eye(dim_z) self._dim_x = dim_x self._dim_z = dim_z self.points_fn = points self._dt = dt self._num_sigmas = points.num_sigmas() self.hx = hx self.fx = fx self.x_mean = x_mean_fn self.z_mean = z_mean_fn # Only computed only if requested via property self._log_likelihood = log(sys.float_info.min) self._likelihood = sys.float_info.min self._mahalanobis = None if sqrt_fn is None: self.msqrt = cholesky else: self.msqrt = sqrt_fn # weights for the means and covariances. self.Wm, self.Wc = points.Wm, points.Wc if residual_x is None: self.residual_x = np.subtract else: self.residual_x = residual_x if residual_z is None: self.residual_z = np.subtract else: self.residual_z = residual_z # 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)) self.sigmas_h = zeros((self._num_sigmas, self._dim_z)) self.K = np.zeros((dim_x, dim_z)) # Kalman gain self.y = np.zeros((dim_z)) # residual self.z = np.array([[None]*dim_z]).T # measurement self.S = np.zeros((dim_z, dim_z)) # system uncertainty self.SI = np.zeros((dim_z, dim_z)) # inverse system uncertainty self.inv = np.linalg.inv # these will always be a copy of x,P after predict() is called self.x_prior = self.x.copy() self.P_prior = self.P.copy() # these will always be a copy of x,P after update() is called self.x_post = self.x.copy() self.P_post = self.P.copy() def predict(self, dt=None, UT=None, fx=None, **fx_args): r""" Performs the predict step of the UKF. On return, self.x and self.P contain the predicted state (x) and covariance (P). ' Important: this MUST be called before update() is called for the first time. Parameters ---------- dt : double, optional If specified, the time step to be used for this prediction. self._dt is used if this is not provided. fx : callable f(x, **fx_args), optional State transition function. If not provided, the default function passed in during construction will be used. 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 - you can use x_mean_fn and z_mean_fn to alter the behavior of the unscented transform. **fx_args : keyword arguments optional keyword arguments to be passed into f(x). """ if dt is None: dt = self._dt if UT is None: UT = unscented_transform # calculate sigma points for given mean and covariance self.compute_process_sigmas(dt, fx, **fx_args) #and pass sigmas through the unscented transform to compute prior self.x, self.P = UT(self.sigmas_f, self.Wm, self.Wc, self.Q, self.x_mean, self.residual_x) # save prior self.x_prior = np.copy(self.x) self.P_prior = np.copy(self.P) def update(self, z, R=None, UT=None, hx=None, **hx_args): """ 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. 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 - you can use x_mean_fn and z_mean_fn to alter the behavior of the unscented transform. **hx_args : keyword argument arguments to be passed into h(x) after x -> h(x, **hx_args) """ if z is None: self.z = np.array([[None]*self._dim_z]).T self.x_post = self.x.copy() self.P_post = self.P.copy() return if hx is None: hx = self.hx if UT is None: UT = unscented_transform if R is None: R = self.R elif isscalar(R): R = eye(self._dim_z) * R # pass prior sigmas through h(x) to get measurement sigmas # the shape of sigmas_h will vary if the shape of z varies, so # recreate each time sigmas_h = [] for s in self.sigmas_f: sigmas_h.append(hx(s, **hx_args)) self.sigmas_h = np.atleast_2d(sigmas_h) # mean and covariance of prediction passed through unscented transform zp, self.S = UT(self.sigmas_h, self.Wm, self.Wc, R, self.z_mean, self.residual_z) self.SI = self.inv(self.S) # compute cross variance of the state and the measurements Pxz = self.cross_variance(self.x, zp, self.sigmas_f, self.sigmas_h) self.K = dot(Pxz, self.SI) # Kalman gain self.y = self.residual_z(z, zp) # residual # update Gaussian state estimate (x, P) self.x = self.x + dot(self.K, self.y) self.P = self.P - dot(self.K, dot(self.S, self.K.T)) # save measurement and posterior state self.z = deepcopy(z) self.x_post = self.x.copy() self.P_post = self.P.copy() # set to None to force recompute self._log_likelihood = None self._likelihood = None self._mahalanobis = None def cross_variance(self, x, z, sigmas_f, sigmas_h): """ Compute cross variance of the state `x` and measurement `z`. """ Pxz = zeros((sigmas_f.shape[1], sigmas_h.shape[1])) N = sigmas_f.shape[0] for i in range(N): dx = self.residual_x(sigmas_f[i], x) dz = self.residual_z(sigmas_h[i], z) Pxz += self.Wc[i] * outer(dx, dz) return Pxz def compute_process_sigmas(self, dt, fx=None, **fx_args): """ computes the values of sigmas_f. Normally a user would not call this, but it is useful if you need to call update more than once between calls to predict (to update for multiple simultaneous measurements), so the sigmas correctly reflect the updated state x, P. """ if fx is None: fx = self.fx # calculate sigma points for given mean and covariance sigmas = self.points_fn.sigma_points(self.x, self.P) for i, s in enumerate(sigmas): self.sigmas_f[i] = fx(s, dt, **fx_args) def batch_filter(self, zs, Rs=None, dts=None, UT=None, saver=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 : None, np.array or list-like, default=None optional list of values to use for the measurement error covariance R. If Rs is None then self.R is used for all epochs. If it is a list of matrices or a 3D array where len(Rs) == len(zs), then it is treated as a list of R values, one per epoch. This allows you to have varying R per epoch. dts : None, scalar or list-like, default=None optional value or list of delta time to be passed into predict. If dtss is None then self.dt is used for all epochs. If it is a list where len(dts) == len(zs), then it is treated as a list of dt values, one per epoch. This allows you to have varying epoch durations. 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 - you can use x_mean_fn and z_mean_fn to alter the behavior of the unscented transform. saver : filterpy.common.Saver, optional filterpy.common.Saver object. If provided, saver.save() will be called after every epoch Returns ------- means: ndarray((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: ndarray((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`. Examples -------- .. code-block:: Python # this example demonstrates tracking a measurement where the time # between measurement varies, as stored in dts The output is then smoothed # with an RTS smoother. zs = [t + random.randn()*4 for t in range (40)] (mu, cov, _, _) = ukf.batch_filter(zs, dts=dts) (xs, Ps, Ks) = ukf.rts_smoother(mu, cov) """ #pylint: disable=too-many-arguments try: z = zs[0] except TypeError: raise TypeError('zs must be list-like') if self._dim_z == 1: if not(isscalar(z) or (z.ndim == 1 and len(z) == 1)): raise TypeError('zs must be a list of scalars or 1D, 1 element arrays') else: if len(z) != self._dim_z: raise TypeError( 'each element in zs must be a 1D array of length {}'.format(self._dim_z)) z_n = np.size(zs, 0) if Rs is None: Rs = [self.R] * z_n if dts is None: dts = [self._dt] * z_n # mean estimates from Kalman Filter if self.x.ndim == 1: means = zeros((z_n, self._dim_x)) else: means = zeros((z_n, self._dim_x, 1)) # state covariances from Kalman Filter covariances = zeros((z_n, self._dim_x, self._dim_x)) for i, (z, r, dt) in enumerate(zip(zs, Rs, dts)): self.predict(dt=dt, UT=UT) self.update(z, r, UT=UT) means[i, :] = self.x covariances[i, :, :] = self.P if saver is not None: saver.save() return (means, covariances) def rts_smoother(self, Xs, Ps, Qs=None, dts=None, UT=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. Qs: 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. 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 - you can use x_mean_fn and z_mean_fn to alter the behavior of the unscented transform. Returns ------- x : numpy.ndarray smoothed means P : numpy.ndarray smoothed state covariances K : numpy.ndarray smoother gain at each step Examples -------- .. code-block:: Python 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) """ #pylint: disable=too-many-locals, too-many-arguments if len(Xs) != len(Ps): raise ValueError('Xs and Ps must have the same length') n, dim_x = Xs.shape if dts is None: dts = [self._dt] * n elif isscalar(dts): dts = [dts] * n if Qs is None: Qs = [self.Q] * n if UT is None: UT = unscented_transform # smoother gain Ks = zeros((n, dim_x, dim_x)) num_sigmas = self._num_sigmas xs, ps = Xs.copy(), Ps.copy() sigmas_f = zeros((num_sigmas, dim_x)) for k in reversed(range(n-1)): # create sigma points from state estimate, pass through state func sigmas = self.points_fn.sigma_points(xs[k], ps[k]) for i in range(num_sigmas): sigmas_f[i] = self.fx(sigmas[i], dts[k]) xb, Pb = UT( sigmas_f, self.Wm, self.Wc, self.Q, self.x_mean, self.residual_x) # compute cross variance Pxb = 0 for i in range(num_sigmas): y = self.residual_x(sigmas_f[i], xb) z = self.residual_x(sigmas[i], Xs[k]) Pxb += self.Wc[i] * outer(z, y) # compute gain K = dot(Pxb, self.inv(Pb)) # update the smoothed estimates xs[k] += dot(K, self.residual_x(xs[k+1], xb)) ps[k] += dot(K, ps[k+1] - Pb).dot(K.T) Ks[k] = K return (xs, ps, Ks) @property def log_likelihood(self): """ log-likelihood of the last measurement. """ if self._log_likelihood is None: self._log_likelihood = logpdf(x=self.y, cov=self.S) return self._log_likelihood @property def likelihood(self): """ Computed from the log-likelihood. The log-likelihood can be very small, meaning a large negative value such as -28000. Taking the exp() of that results in 0.0, which can break typical algorithms which multiply by this value, so by default we always return a number >= sys.float_info.min. """ if self._likelihood is None: self._likelihood = exp(self.log_likelihood) if self._likelihood == 0: self._likelihood = sys.float_info.min return self._likelihood @property def mahalanobis(self): """" Mahalanobis distance of measurement. E.g. 3 means measurement was 3 standard deviations away from the predicted value. Returns ------- mahalanobis : float """ if self._mahalanobis is None: self._mahalanobis = sqrt(float(dot(dot(self.y.T, self.SI), self.y))) return self._mahalanobis def __repr__(self): return '\n'.join([ 'UnscentedKalmanFilter object', pretty_str('x', self.x), pretty_str('P', self.P), pretty_str('x_prior', self.x_prior), pretty_str('P_prior', self.P_prior), pretty_str('Q', self.Q), pretty_str('R', self.R), pretty_str('S', self.S), pretty_str('K', self.K), pretty_str('y', self.y), pretty_str('log-likelihood', self.log_likelihood), pretty_str('likelihood', self.likelihood), pretty_str('mahalanobis', self.mahalanobis), pretty_str('sigmas_f', self.sigmas_f), pretty_str('h', self.sigmas_h), pretty_str('Wm', self.Wm), pretty_str('Wc', self.Wc), pretty_str('residual_x', self.residual_x), pretty_str('residual_z', self.residual_z), pretty_str('msqrt', self.msqrt), pretty_str('hx', self.hx), pretty_str('fx', self.fx), pretty_str('x_mean', self.x_mean), pretty_str('z_mean', self.z_mean) ])