248 lines
6.9 KiB
Python
248 lines
6.9 KiB
Python
# -*- coding: utf-8 -*-
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"""Copyright 2015 Roger R Labbe Jr.
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Code supporting the book
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Kalman and Bayesian Filters in Python
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https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python
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This is licensed under an MIT license. See the LICENSE.txt file
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for more information.
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"""
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from __future__ import (absolute_import, division, print_function,
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unicode_literals)
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import book_plots as bp
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import filterpy.kalman as kf
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from math import radians, sin, cos, sqrt, exp
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import matplotlib.pyplot as plt
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import numpy as np
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import numpy.random as random
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def ball_kf(x, y, omega, v0, dt, r=0.5, q=0.02):
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g = 9.8 # gravitational constant
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f1 = kf.KalmanFilter(dim_x=5, dim_z=2)
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ay = .5*dt**2
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f1.F = np.array ([[1, dt, 0, 0, 0], # x = x0+dx*dt
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[0, 1, 0, 0, 0], # dx = dx
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[0, 0, 1, dt, ay], # y = y0 +dy*dt+1/2*g*dt^2
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[0, 0, 0, 1, dt], # dy = dy0 + ddy*dt
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[0, 0, 0, 0, 1]]) # ddy = -g.
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f1.H = np.array([
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[1, 0, 0, 0, 0],
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[0, 0, 1, 0, 0]])
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f1.R *= r
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f1.Q *= q
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omega = radians(omega)
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vx = cos(omega) * v0
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vy = sin(omega) * v0
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f1.x = np.array([[x,vx,y,vy,-9.8]]).T
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return f1
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def plot_radar(xs, track, time):
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plt.figure()
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bp.plot_track(time, track[:, 0])
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bp.plot_filter(time, xs[:, 0])
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plt.legend(loc=4)
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plt.xlabel('time (sec)')
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plt.ylabel('position (m)')
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plt.figure()
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bp.plot_track(time, track[:, 1])
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bp.plot_filter(time, xs[:, 1])
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plt.legend(loc=4)
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plt.xlabel('time (sec)')
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plt.ylabel('velocity (m/s)')
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plt.figure()
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bp.plot_track(time, track[:, 2])
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bp.plot_filter(time, xs[:, 2])
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plt.ylabel('altitude (m)')
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plt.legend(loc=4)
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plt.xlabel('time (sec)')
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plt.ylim((900, 1600))
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plt.show()
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def plot_bicycle():
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plt.clf()
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plt.axes()
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ax = plt.gca()
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#ax.add_patch(plt.Rectangle((10,0), 10, 20, fill=False, ec='k')) #car
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ax.add_patch(plt.Rectangle((21,1), .75, 2, fill=False, ec='k')) #wheel
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ax.add_patch(plt.Rectangle((21.33,10), .75, 2, fill=False, ec='k', angle=20)) #wheel
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ax.add_patch(plt.Rectangle((21.,4.), .75, 2, fill=True, ec='k', angle=5, ls='dashdot', alpha=0.3)) #wheel
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plt.arrow(0, 2, 20.5, 0, fc='k', ec='k', head_width=0.5, head_length=0.5)
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plt.arrow(0, 2, 20.4, 3, fc='k', ec='k', head_width=0.5, head_length=0.5)
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plt.arrow(21.375, 2., 0, 8.5, fc='k', ec='k', head_width=0.5, head_length=0.5)
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plt.arrow(23, 2, 0, 2.5, fc='k', ec='k', head_width=0.5, head_length=0.5)
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#ax.add_patch(plt.Rectangle((10,0), 10, 20, fill=False, ec='k'))
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plt.text(11, 1.0, "R", fontsize=18)
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plt.text(8, 2.2, r"$\beta$", fontsize=18)
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plt.text(20.4, 13.5, r"$\alpha$", fontsize=18)
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plt.text(21.6, 7, "w (wheelbase)", fontsize=18)
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plt.text(0, 1, "C", fontsize=18)
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plt.text(24, 3, "d", fontsize=18)
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plt.plot([21.375, 21.25], [11, 14], color='k', lw=1)
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plt.plot([21.375, 20.2], [11, 14], color='k', lw=1)
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plt.axis('scaled')
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plt.xlim(0,25)
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plt.axis('off')
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plt.show()
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#plot_bicycle()
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class BaseballPath(object):
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def __init__(self, x0, y0, launch_angle_deg, velocity_ms, noise=(1.0,1.0)):
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""" Create baseball path object in 2D (y=height above ground)
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x0,y0 initial position
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launch_angle_deg angle ball is travelling respective to ground plane
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velocity_ms speeed of ball in meters/second
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noise amount of noise to add to each reported position in (x,y)
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"""
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omega = radians(launch_angle_deg)
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self.v_x = velocity_ms * cos(omega)
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self.v_y = velocity_ms * sin(omega)
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self.x = x0
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self.y = y0
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self.noise = noise
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def drag_force (self, velocity):
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""" Returns the force on a baseball due to air drag at
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the specified velocity. Units are SI
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"""
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B_m = 0.0039 + 0.0058 / (1. + exp((velocity-35.)/5.))
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return B_m * velocity
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def update(self, dt, vel_wind=0.):
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""" compute the ball position based on the specified time step and
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wind velocity. Returns (x,y) position tuple.
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"""
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# Euler equations for x and y
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self.x += self.v_x*dt
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self.y += self.v_y*dt
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# force due to air drag
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v_x_wind = self.v_x - vel_wind
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v = sqrt (v_x_wind**2 + self.v_y**2)
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F = self.drag_force(v)
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# Euler's equations for velocity
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self.v_x = self.v_x - F*v_x_wind*dt
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self.v_y = self.v_y - 9.81*dt - F*self.v_y*dt
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return (self.x + random.randn()*self.noise[0],
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self.y + random.randn()*self.noise[1])
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def plot_ball():
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y = 1.
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x = 0.
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theta = 35. # launch angle
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v0 = 50.
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dt = 1/10. # time step
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ball = BaseballPath(x0=x, y0=y, launch_angle_deg=theta, velocity_ms=v0, noise=[.3,.3])
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f1 = ball_kf(x,y,theta,v0,dt,r=1.)
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f2 = ball_kf(x,y,theta,v0,dt,r=10.)
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t = 0
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xs = []
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ys = []
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xs2 = []
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ys2 = []
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while f1.x[2,0] > 0:
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t += dt
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x,y = ball.update(dt)
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z = np.mat([[x,y]]).T
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f1.update(z)
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f2.update(z)
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xs.append(f1.x[0,0])
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ys.append(f1.x[2,0])
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xs2.append(f2.x[0,0])
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ys2.append(f2.x[2,0])
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f1.predict()
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f2.predict()
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p1 = plt.scatter(x, y, color='green', marker='o', s=75, alpha=0.5)
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p2, = plt.plot (xs, ys,lw=2)
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p3, = plt.plot (xs2, ys2,lw=4, c='r')
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plt.legend([p1,p2, p3], ['Measurements', 'Kalman filter(R=0.5)', 'Kalman filter(R=10)'])
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plt.show()
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def show_radar_chart():
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plt.xlim([0.9,2.5])
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plt.ylim([0.5,2.5])
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plt.scatter ([1,2],[1,2])
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#plt.scatter ([2],[1],marker='o')
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ax = plt.axes()
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ax.annotate('', xy=(2,2), xytext=(1,1),
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arrowprops=dict(arrowstyle='->', ec='r',shrinkA=3, shrinkB=4))
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ax.annotate('', xy=(2,1), xytext=(1,1),
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arrowprops=dict(arrowstyle='->', ec='b',shrinkA=0, shrinkB=0))
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ax.annotate('', xy=(2,2), xytext=(2,1),
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arrowprops=dict(arrowstyle='->', ec='b',shrinkA=0, shrinkB=4))
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ax.annotate('$\Theta$', xy=(1.2, 1.05), color='b')
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ax.annotate('Aircraft', xy=(2.04,2.), color='b')
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ax.annotate('altitude (y)', xy=(2.04,1.5), color='k')
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ax.annotate('x', xy=(1.5, .9))
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ax.annotate('Radar', xy=(.95, 0.8))
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ax.annotate('Slant\n (r)', xy=(1.5,1.62), color='r')
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plt.title("Radar Tracking")
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ax.xaxis.set_ticklabels([])
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ax.yaxis.set_ticklabels([])
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ax.xaxis.set_ticks([])
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ax.yaxis.set_ticks([])
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plt.show()
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def show_linearization():
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xs = np.arange(0, 2, 0.01)
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ys = [x**2 - 2*x for x in xs]
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def y(x):
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return x - 2.25
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plt.plot(xs, ys, label='$f(x)=x^2−2x$')
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plt.plot([1, 2], [y(1), y(2)], color='k', ls='--', label='linearization')
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plt.axes().axvline(1.5, lw=1, c='k')
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plt.xlim(0, 2)
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plt.ylim([-1.5, 0.0])
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plt.title('Linearization of $f(x)$ at $x=1.5$')
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plt.xlabel('$x=1.5$')
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plt.legend(loc=4)
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plt.show() |