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ensemble_kalman_filter.py
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ensemble_kalman_filter.py
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"""
Ensemble Kalman Filter(EnKF) localization sample
author: Ryohei Sasaki(rsasaki0109)
Ref:
- [Ensemble Kalman filtering](https://rmets.onlinelibrary.wiley.com/doi/10.1256/qj.05.135)
"""
import numpy as np
import math
import matplotlib.pyplot as plt
# Simulation parameter
Qsim = np.diag([0.2, np.deg2rad(1.0)])**2
Rsim = np.diag([1.0, np.deg2rad(30.0)])**2
DT = 0.1 # time tick [s]
SIM_TIME = 50.0 # simulation time [s]
MAX_RANGE = 20.0 # maximum observation range
# Ensemble Kalman filter parameter
NP = 20 # Number of Particle
show_animation = True
def calc_input():
v = 1.0 # [m/s]
yawrate = 0.1 # [rad/s]
u = np.array([[v, yawrate]]).T
return u
def observation(xTrue, xd, u, RFID):
xTrue = motion_model(xTrue, u)
z = np.zeros((0, 3))
for i in range(len(RFID[:, 0])):
dx = RFID[i, 0] - xTrue[0, 0]
dy = RFID[i, 1] - xTrue[1, 0]
d = math.sqrt(dx**2 + dy**2)
if d <= MAX_RANGE:
dn = d + np.random.randn() * Qsim[0, 0] # add noise
zi = np.array([dn, RFID[i, 0], RFID[i, 1]])
z = np.vstack((z, zi))
# add noise to input
ud = np.array([[
u[0, 0] + np.random.randn() * Rsim[0, 0],
u[1, 0] + np.random.randn() * Rsim[1, 1]]]).T
xd = motion_model(xd, ud)
return xTrue, z, xd, ud
def motion_model(x, u):
F = np.array([[1.0, 0, 0, 0],
[0, 1.0, 0, 0],
[0, 0, 1.0, 0],
[0, 0, 0, 0]])
B = np.array([[DT * math.cos(x[2, 0]), 0],
[DT * math.sin(x[2, 0]), 0],
[0.0, DT],
[1.0, 0.0]])
x = F.dot(x) + B.dot(u)
return x
def calc_covariance(xEst, px):
cov = np.zeros((3, 3))
for i in range(px.shape[1]):
dx = (px[:, i] - xEst)[0:3]
cov += dx.dot(dx.T)
return cov
def enkf_localization(px, xEst, PEst, z, u):
"""
Localization with Ensemble Kalman filter
"""
pz = np.zeros((z.shape[0], NP)) # Particle store of z
for ip in range(NP):
x = np.array([px[:, ip]]).T
# Predict with random input sampling
ud1 = u[0, 0] + np.random.randn() * Rsim[0, 0]
ud2 = u[1, 0] + np.random.randn() * Rsim[1, 1]
ud = np.array([[ud1, ud2]]).T
x = motion_model(x, ud)
px[:, ip] = x[:, 0]
for i in range(len(z[:, 0])):
dx = x[0, 0] - z[i, 1]
dy = x[1, 0] - z[i, 2]
prez = math.sqrt(dx**2 + dy**2) + np.random.randn() * Qsim[0, 0] # add noise
pz[i, ip] = prez
x_ave = np.mean(px, axis=1)
x_dif = px - np.tile(x_ave, (NP, 1)).T
z_ave = np.mean(pz, axis=1)
z_dif = pz - np.tile(z_ave, (NP, 1)).T
U = 1/(NP-1) * x_dif @ z_dif.T
V = 1/(NP-1) * z_dif @ z_dif.T
K = U @ np.linalg.inv(V) # Kalman Gain
px_hat = px + K @ (np.tile(z[:,0], (NP, 1)).T - pz)
xEst = np.average(px_hat, axis=1).reshape(4, 1)
PEst = calc_covariance(xEst, px_hat)
return xEst, PEst, px_hat
def plot_covariance_ellipse(xEst, PEst): # pragma: no cover
Pxy = PEst[0:2, 0:2]
eigval, eigvec = np.linalg.eig(Pxy)
if eigval[0] >= eigval[1]:
bigind = 0
smallind = 1
else:
bigind = 1
smallind = 0
t = np.arange(0, 2 * math.pi + 0.1, 0.1)
# eigval[bigind] or eiqval[smallind] were occassionally negative numbers extremely
# close to 0 (~10^-20), catch these cases and set the respective variable to 0
try:
a = math.sqrt(eigval[bigind])
except ValueError:
a = 0
try:
b = math.sqrt(eigval[smallind])
except ValueError:
b = 0
x = [a * math.cos(it) for it in t]
y = [b * math.sin(it) for it in t]
angle = math.atan2(eigvec[bigind, 1], eigvec[bigind, 0])
R = np.array([[math.cos(angle), math.sin(angle)],
[-math.sin(angle), math.cos(angle)]])
fx = R.dot(np.array([[x, y]]))
px = np.array(fx[0, :] + xEst[0, 0]).flatten()
py = np.array(fx[1, :] + xEst[1, 0]).flatten()
plt.plot(px, py, "--r")
def main():
print(__file__ + " start!!")
time = 0.0
# RFID positions [x, y]
RFID = np.array([[10.0, 0.0],
[10.0, 10.0],
[0.0, 15.0],
[-5.0, 20.0]])
# State Vector [x y yaw v]'
xEst = np.zeros((4, 1))
xTrue = np.zeros((4, 1))
PEst = np.eye(4)
px = np.zeros((4, NP)) # Particle store of x
xDR = np.zeros((4, 1)) # Dead reckoning
# history
hxEst = xEst
hxTrue = xTrue
hxDR = xTrue
while SIM_TIME >= time:
time += DT
u = calc_input()
xTrue, z, xDR, ud = observation(xTrue, xDR, u, RFID)
xEst, PEst, px = enkf_localization(px, xEst, PEst, z, ud)
# store data history
hxEst = np.hstack((hxEst, xEst))
hxDR = np.hstack((hxDR, xDR))
hxTrue = np.hstack((hxTrue, xTrue))
if show_animation:
plt.cla()
for i in range(len(z[:, 0])):
plt.plot([xTrue[0, 0], z[i, 1]], [xTrue[1, 0], z[i, 2]], "-k")
plt.plot(RFID[:, 0], RFID[:, 1], "*k")
plt.plot(px[0, :], px[1, :], ".r")
plt.plot(np.array(hxTrue[0, :]).flatten(),
np.array(hxTrue[1, :]).flatten(), "-b")
plt.plot(np.array(hxDR[0, :]).flatten(),
np.array(hxDR[1, :]).flatten(), "-k")
plt.plot(np.array(hxEst[0, :]).flatten(),
np.array(hxEst[1, :]).flatten(), "-r")
#plot_covariance_ellipse(xEst, PEst)
plt.axis("equal")
plt.grid(True)
plt.pause(0.001)
if __name__ == '__main__':
main()