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Extended Kalman filter sample
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# -*- coding: utf-8 -*- | |
import numpy as np | |
import matplotlib.pyplot as plt | |
def main(): | |
# 初期化 | |
T = 30 # 観測数 | |
r = 10.0 # 半径 | |
w = 1.0*10/180 * np.pi # 角速度[rad/s] | |
x = np.mat([[0],[-5]]) # 初期位置 | |
X = [x] # 実際の状態 | |
Y = [x] # 観測 | |
U = np.mat([[r],[w]]) # 操作量(一定) | |
# state x = f(x_,u,v), v~N(0,Q) | |
Q = np.mat([[0.5,0.0],[0.0,0.5]]) | |
# observation Y = x + w, w~N(0,R) | |
R = np.mat([[0.5,0.0], [0.0,0.5]]) | |
def f(t,x,u): | |
x0 = u[0,0]*u[1,0]*np.cos(u[1,0]*t)+x[0,0] | |
x1 = u[0,0]*u[1,0]*np.sin(u[1,0]*t)+x[1,0] | |
return np.mat([[x0],[x1]]) | |
def Jf(t,x,u): | |
""" | |
解析的に求めるf(x)のヤコビ行列 | |
""" | |
return np.mat([[u[0,0]*u[1,0]*np.cos(u[1,0]*t),0],[0,u[0,0]*u[1,0]*np.sin(u[1,0]*t)]]) | |
# 観測データの生成 | |
for t in range(T): | |
x = f(t,x,U)+np.random.multivariate_normal([0,0],Q,1).T | |
X.append(x) | |
y = x + np.random.multivariate_normal([0,0],R,1).T | |
Y.append(y) | |
# EKF | |
_x = np.mat([[0],[-5]]) | |
Sigma = np.mat([[1,0],[0,1]]) | |
_X = [_x] # 推定 | |
for t in range(T): | |
# prediction | |
A = Jf(t,_x,U) | |
_x_ = f(t,_x,U) | |
Sigma_ = Q + A * Sigma * A.T | |
# update | |
yi = Y[t+1] - _x_ | |
S = Sigma_ + R | |
G = Sigma_ * S.I | |
_x = _x_ + G * yi | |
Sigma = Sigma_ - G * Sigma_ | |
_X.append(_x) | |
# 描画 | |
a, b = np.array(np.concatenate(X,axis=1)) | |
plt.plot(a,b,'rs-', label="X_correct") | |
a, b = np.array(np.concatenate(Y,axis=1)) | |
plt.plot(a,b,'g^-', label="Y (=X+N(0,R))") | |
a, b = np.array(np.concatenate(_X,axis=1)) | |
plt.plot(a,b,'bo-', label="X_estimate") | |
plt.legend(bbox_to_anchor=(0.80, 0.00), loc='lower left', borderaxespad=0) | |
plt.axis('equal') | |
plt.show() | |
if __name__ == '__main__': | |
main() |
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Sample of extended Kalman filter (EKF)