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ukf_internal.py
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/
ukf_internal.py
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# -*- coding: utf-8 -*-
"""Copyright 2015 Roger R Labbe Jr.
Code supporting the book
Kalman and Bayesian Filters in Python
https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python
This is licensed under an MIT license. See the LICENSE.txt file
for more information.
"""
from __future__ import (absolute_import, division, print_function,
unicode_literals)
from filterpy.kalman import UnscentedKalmanFilter as UKF
from filterpy.kalman import MerweScaledSigmaPoints
import filterpy.stats as stats
from filterpy.stats import plot_covariance_ellipse
import matplotlib.pyplot as plt
from matplotlib.patches import Ellipse,Arrow
import math
from math import cos, sin, atan2, pi
import numpy as np
from numpy.random import randn
def _sigma_points(mean, sigma, kappa):
sigma1 = mean + math.sqrt((1+kappa)*sigma)
sigma2 = mean - math.sqrt((1+kappa)*sigma)
return mean, sigma1, sigma2
def arrow(x1,y1,x2,y2, width=0.2):
return Arrow(x1,y1, x2-x1, y2-y1, lw=1, width=width, ec='k', color='k')
def show_two_sensor_bearing():
circle1=plt.Circle((-4,0),5,color='#004080',fill=False,linewidth=20, alpha=.7)
circle2=plt.Circle((4,0),5,color='#E24A33', fill=False, linewidth=5, alpha=.7)
fig = plt.gcf()
ax = fig.gca()
plt.axis('equal')
#plt.xlim((-10,10))
plt.ylim((-6,6))
plt.plot ([-4,0], [0,3], c='#004080')
plt.plot ([4,0], [0,3], c='#E24A33')
plt.text(-4, -.5, "A", fontsize=16, horizontalalignment='center')
plt.text(4, -.5, "B", fontsize=16, horizontalalignment='center')
ax.add_patch(circle1)
ax.add_patch(circle2)
plt.show()
def show_three_gps():
circle1=plt.Circle((-4,0),5,color='#004080',fill=False,linewidth=20, alpha=.7)
circle2=plt.Circle((4,0),5,color='#E24A33', fill=False, linewidth=8, alpha=.7)
circle3=plt.Circle((0,-3),6,color='#534543',fill=False, linewidth=13, alpha=.7)
fig = plt.gcf()
ax = fig.gca()
ax.add_patch(circle1)
ax.add_patch(circle2)
ax.add_patch(circle3)
plt.axis('equal')
plt.show()
def show_four_gps():
circle1=plt.Circle((-4,2),5,color='#004080',fill=False,linewidth=20, alpha=.7)
circle2=plt.Circle((5.5,1),5,color='#E24A33', fill=False, linewidth=8, alpha=.7)
circle3=plt.Circle((0,-3),6,color='#534543',fill=False, linewidth=13, alpha=.7)
circle4=plt.Circle((0,8),5,color='#214513',fill=False, linewidth=13, alpha=.7)
fig = plt.gcf()
ax = fig.gca()
ax.add_patch(circle1)
ax.add_patch(circle2)
ax.add_patch(circle3)
ax.add_patch(circle4)
plt.axis('equal')
plt.show()
def show_sigma_transform(with_text=False):
fig = plt.gcf()
ax=fig.gca()
x = np.array([0, 5])
P = np.array([[4, -2.2], [-2.2, 3]])
plot_covariance_ellipse(x, P, facecolor='b', alpha=0.6, variance=9)
sigmas = MerweScaledSigmaPoints(2, alpha=.5, beta=2., kappa=0.)
S = sigmas.sigma_points(x=x, P=P)
plt.scatter(S[:,0], S[:,1], c='k', s=80)
x = np.array([15, 5])
P = np.array([[3, 1.2],[1.2, 6]])
plot_covariance_ellipse(x, P, facecolor='g', variance=9, alpha=0.3)
ax.add_artist(arrow(S[0,0], S[0,1], 11, 4.1, 0.6))
ax.add_artist(arrow(S[1,0], S[1,1], 13, 7.7, 0.6))
ax.add_artist(arrow(S[2,0], S[2,1], 16.3, 0.93, 0.6))
ax.add_artist(arrow(S[3,0], S[3,1], 16.7, 10.8, 0.6))
ax.add_artist(arrow(S[4,0], S[4,1], 17.7, 5.6, 0.6))
ax.axes.get_xaxis().set_visible(False)
ax.axes.get_yaxis().set_visible(False)
if with_text:
plt.text(2.5, 1.5, r"$\chi$", fontsize=32)
plt.text(13, -1, r"$\mathcal{Y}$", fontsize=32)
#plt.axis('equal')
plt.show()
def show_2d_transform():
plt.cla()
ax=plt.gca()
ax.add_artist(Ellipse(xy=(2,5), width=2, height=3,angle=70,linewidth=1,ec='k'))
ax.add_artist(Ellipse(xy=(7,5), width=2.2, alpha=0.3, height=3.8,angle=150,fc='g',linewidth=1,ec='k'))
ax.add_artist(arrow(2, 5, 6, 4.8))
ax.add_artist(arrow(1.5, 5.5, 7, 3.8))
ax.add_artist(arrow(2.3, 4.1, 8, 6))
ax.add_artist(arrow(3.3, 5.1, 6.5, 4.3))
ax.add_artist(arrow(1.3, 4.8, 7.2, 6.3))
ax.add_artist(arrow(1.1, 5.2, 8.2, 5.3))
ax.add_artist(arrow(2, 4.4, 7.3, 4.5))
ax.axes.get_xaxis().set_visible(False)
ax.axes.get_yaxis().set_visible(False)
plt.axis('equal')
plt.xlim(0,10); plt.ylim(0,10)
plt.show()
def show_3_sigma_points():
xs = np.arange(-4, 4, 0.1)
var = 1.5
ys = [stats.gaussian(x, 0, var) for x in xs]
samples = [0, 1.2, -1.2]
for x in samples:
plt.scatter ([x], [stats.gaussian(x, 0, var)], s=80)
plt.plot(xs, ys)
plt.show()
def _plot_sigmas(s, w, alpha=0.5, **kwargs):
min_w = min(abs(w))
scale_factor = 100 / min_w
return plt.scatter(s[:, 0], s[:, 1], s=abs(w)*scale_factor,
alpha=alpha, **kwargs)
def show_sigma_selections():
ax=plt.gca()
ax.axes.get_xaxis().set_visible(False)
ax.axes.get_yaxis().set_visible(False)
x = np.array([2, 5])
P = np.array([[3, 1.1], [1.1, 4]])
points = MerweScaledSigmaPoints(2, .09, 2., 1.)
sigmas = points.sigma_points(x, P)
Wm, Wc = points.Wm, points.Wc
plot_covariance_ellipse(x, P, facecolor='b', alpha=.3, variance=[.5])
_plot_sigmas(sigmas, Wc, alpha=1.0, facecolor='k')
x = np.array([5, 5])
points = MerweScaledSigmaPoints(2, .15, 1., .15)
sigmas = points.sigma_points(x, P)
Wm, Wc = points.Wm, points.Wc
plot_covariance_ellipse(x, P, facecolor='b', alpha=0.3, variance=[.5])
_plot_sigmas(sigmas, Wc, alpha=1.0, facecolor='k')
x = np.array([8, 5])
points = MerweScaledSigmaPoints(2, .2, 3., 10)
sigmas = points.sigma_points(x, P)
Wm, Wc = points.Wm, points.Wc
plot_covariance_ellipse(x, P, facecolor='b', alpha=0.3, variance=[.5])
_plot_sigmas(sigmas, Wc, alpha=1.0, facecolor='k')
plt.axis('equal')
plt.xlim(0,10); plt.ylim(0,10)
plt.show()
def show_sigmas_for_2_kappas():
# generate the Gaussian data
xs = np.arange(-4, 4, 0.1)
mean = 0
sigma = 1.5
ys = [stats.gaussian(x, mean, sigma*sigma) for x in xs]
#generate our samples
kappa = 2
x0,x1,x2 = _sigma_points(mean, sigma, kappa)
samples = [x0,x1,x2]
for x in samples:
p1 = plt.scatter([x], [stats.gaussian(x, mean, sigma*sigma)], s=80, color='k')
kappa = -.5
x0,x1,x2 = _sigma_points(mean, sigma, kappa)
samples = [x0,x1,x2]
for x in samples:
p2 = plt.scatter([x], [stats.gaussian(x, mean, sigma*sigma)], s=80, color='b')
plt.legend([p1,p2], ['$kappa$=2', '$kappa$=-0.5'])
plt.plot(xs, ys)
plt.show()
def plot_sigmas(sigmas, x, cov):
if not np.isscalar(cov):
cov = np.atleast_2d(cov)
pts = sigmas.sigma_points(x=x, P=cov)
plt.scatter(pts[:, 0], pts[:, 1], s=sigmas.Wm*1000)
plt.axis('equal')
def plot_sigma_points():
x = np.array([0, 0])
P = np.array([[4, 2], [2, 4]])
sigmas = MerweScaledSigmaPoints(n=2, alpha=.3, beta=2., kappa=1.)
S0 = sigmas.sigma_points(x, P)
Wm0, Wc0 = sigmas.Wm, sigmas.Wc
sigmas = MerweScaledSigmaPoints(n=2, alpha=1., beta=2., kappa=1.)
S1 = sigmas.sigma_points(x, P)
Wm1, Wc1 = sigmas.Wm, sigmas.Wc
def plot_sigmas(s, w, **kwargs):
min_w = min(abs(w))
scale_factor = 100 / min_w
return plt.scatter(s[:, 0], s[:, 1], s=abs(w)*scale_factor, alpha=.5, **kwargs)
plt.subplot(121)
plot_sigmas(S0, Wc0, c='b')
plot_covariance_ellipse(x, P, facecolor='g', alpha=0.2, variance=[1, 4])
plt.title('alpha=0.3')
plt.subplot(122)
plot_sigmas(S1, Wc1, c='b', label='Kappa=2')
plot_covariance_ellipse(x, P, facecolor='g', alpha=0.2, variance=[1, 4])
plt.title('alpha=1')
plt.show()
def plot_radar(xs, t, plot_x=True, plot_vel=True, plot_alt=True):
xs = np.asarray(xs)
if plot_x:
plt.figure()
plt.plot(t, xs[:, 0]/1000.)
plt.xlabel('time(sec)')
plt.ylabel('position(km)')
plt.tight_layout()
if plot_vel:
plt.figure()
plt.plot(t, xs[:, 1])
plt.xlabel('time(sec)')
plt.ylabel('velocity')
plt.tight_layout()
if plot_alt:
plt.figure()
plt.plot(t, xs[:,2])
plt.xlabel('time(sec)')
plt.ylabel('altitude')
plt.tight_layout()
plt.show()
def plot_altitude(xs, t, track):
xs = np.asarray(xs)
plt.plot(t, xs[:,2], label='filter', )
plt.plot(t, track, label='Aircraft', lw=2, ls='--', c='k')
plt.xlabel('time(sec)')
plt.ylabel('altitude')
plt.legend(loc=4)
def print_sigmas(n=1, mean=5, cov=3, alpha=.1, beta=2., kappa=2):
points = MerweScaledSigmaPoints(n, alpha, beta, kappa)
print('sigmas: ', points.sigma_points(mean, cov).T[0])
Wm, Wc = points.Wm, points.Wc
print('mean weights:', Wm)
print('cov weights:', Wc)
print('lambda:', alpha**2 *(n+kappa) - n)
print('sum cov', sum(Wc))
def plot_rts_output(xs, Ms, t):
plt.figure()
plt.plot(t, xs[:, 0]/1000., label='KF', lw=2)
plt.plot(t, Ms[:, 0]/1000., c='k', label='RTS', lw=2)
plt.xlabel('time(sec)')
plt.ylabel('x')
plt.legend(loc=4)
plt.figure()
plt.plot(t, xs[:, 1], label='KF')
plt.plot(t, Ms[:, 1], c='k', label='RTS')
plt.xlabel('time(sec)')
plt.ylabel('x velocity')
plt.legend(loc=4)
plt.figure()
plt.plot(t, xs[:, 2], label='KF')
plt.plot(t, Ms[:, 2], c='k', label='RTS')
plt.xlabel('time(sec)')
plt.ylabel('Altitude(m)')
plt.legend(loc=4)
np.set_printoptions(precision=4)
print('Difference in position in meters:\n\t', xs[-6:-1, 0] - Ms[-6:-1, 0])
def plot_scatter_of_bearing_error():
def plot_scatter(theta):
theta = math.radians(theta)
d = 100
xs, ys = [], []
for i in range (3000):
a = theta + randn() * math.radians(1)
xs.append(d*math.cos(a))
ys.append(d*math.sin(a))
plt.scatter(xs, ys)
plt.xlabel('x')
plt.ylabel('y')
plt.subplot(121)
plot_scatter(45)
plt.gca().set_aspect('equal')
plt.title("45° bearing")
plt.subplot(122)
plot_scatter(180)
plt.xlim((-101, -99))
plt.title("180° bearing")
def plot_scatter_moving_target():
pos = np.array([5., 5.])
for i in range(5):
pos += (0.5, 1.)
actual_angle = math.atan2(pos[1], pos[0])
d = math.sqrt(pos[0]**2 + pos[1]**2)
xs, ys = [], []
for i in range (100):
a = actual_angle + randn() * math.radians(1)
xs.append(d*math.cos(a))
ys.append(d*math.sin(a))
plt.scatter(xs, ys, c='C0')
plt.axis('equal')
plt.plot([5.5, pos[0]], [6, pos[1]], c='g', linestyle='--')
def _isct(pa, pb, alpha, beta):
""" Returns the (x, y) intersections of points pa and pb
given the bearing ba for point pa and bearing bb for
point pb.
"""
B = [pb[0] - pa[0], pb[1] - pa[1]]
AB = math.sqrt((pa[0] - pb[0])**2 + (pa[1] - pb[1])**2)
ab = atan2(B[1], B[0])
a = alpha - ab
b = pi - beta - ab
p = pi - b - a
AP = (sin(b) / sin(p)) * AB
x = cos(alpha) * AP + pa[0]
y = sin(alpha) * AP + pa[1]
return x, y
def _plot_iscts(pos, sa, sb, N=4):
for i in range(N):
pos += (0.5, 1.)
actual_angle_a = math.atan2(pos[1] - sa[1], pos[0] - sa[0])
actual_angle_b = math.atan2(pos[1] - sb[1], pos[0] - sb[0])
da = math.sqrt((sa[0] - pos[0])**2 + (sa[1] - pos[1])**2)
db = math.sqrt((sb[0] - pos[0])**2 + (sb[1] - pos[1])**2)
xs, ys, xs_a, xs_b, ys_a, ys_b = [], [], [], [], [], []
for i in range (300):
a_a = actual_angle_a + randn() * math.radians(1)
a_b = actual_angle_b + randn() * math.radians(1)
x,y = _isct(sa, sb, a_a, a_b)
xs.append(x)
ys.append(y)
xs_a.append(da*math.cos(a_a) + sa[0])
ys_a.append(da*math.sin(a_a) + sa[1])
xs_b.append(db*math.cos(a_b) + sb[0])
ys_b.append(db*math.sin(a_b) + sb[1])
plt.scatter(xs, ys, c='r', marker='.', alpha=0.5)
plt.scatter(xs_a, ys_a, c='k', edgecolor='k')
plt.scatter(xs_b, ys_b, marker='v', edgecolor=None, c='C0')
plt.gca().set_aspect('equal')
def plot_iscts_two_sensors():
plt.subplot(121)
pos = np.array([4., 4,])
sa = [0., 2.]
sb = [8., 2.]
plt.scatter(*sa, s=200, c='k', marker='v')
plt.scatter(*sb, s=200, marker='s', c='C0')
_plot_iscts(pos, sa, sb, N=4)
plt.subplot(122)
plot_iscts_two_sensors_changed_sensors()
def plot_iscts_two_sensors_changed_sensors():
sa = [3, 4]
sb = [3, 7]
pos= np.array([3., 3.])
plt.scatter(*sa, s=200, c='k', marker='v')
plt.scatter(*sb, s=200, marker='s')
_plot_iscts(pos, sa, sb, N=5)
plt.ylim(3.8, 8.5)
if __name__ == '__main__':
#show_2d_transform()
#show_sigma_selections()
plot_scatter_of_bearing_error()
#show_sigma_transform(True)
#show_four_gps()
#show_sigma_transform()
#show_sigma_selections()