From ef900b344578249aab915625e33d991865cc3d08 Mon Sep 17 00:00:00 2001
From: DaMichel <mw.pub@welter-4d.de>
Date: Sat, 9 Jul 2016 10:38:12 +0200
Subject: simple sim in python for playing with kalman filtering

---
 filter-kalman/kalman_simulation.py | 76 ++++++++++++++++++++++++++++++++++++++
 1 file changed, 76 insertions(+)
 create mode 100644 filter-kalman/kalman_simulation.py

diff --git a/filter-kalman/kalman_simulation.py b/filter-kalman/kalman_simulation.py
new file mode 100644
index 00000000..7a320426
--- /dev/null
+++ b/filter-kalman/kalman_simulation.py
@@ -0,0 +1,76 @@
+# -*- coding: utf-8 -*-
+import numpy as np
+import numpy.matlib as mt
+import matplotlib.pyplot as plt
+
+M = 2
+N = 500
+dt = 1.0 / 30.
+sigma_measure = 0.1
+# x = 1/2 a t**2
+# assume x = 0.5 in 0.5, i.e. head turns half maximal movement range in 0.5 sec.
+# so a = 2 x / t**2 =  4.0
+sigma_accel   = 4.0 
+
+x = np.zeros(N, dtype = np.float)
+x[N/3:] = 1.0 # step input
+measurement = x.copy()
+measurement += np.random.normal(0., sigma_measure, N)
+
+A = np.matrix([
+    [ 1. , dt ],
+    [ 0. , 1. ]
+])
+
+R = np.matrix([[ sigma_measure**2 ]])
+
+dv = sigma_accel * dt
+dp = sigma_accel * 0.5 * dt * dt
+Q = np.matrix([
+    [ dp*dp, dp*dv ],
+    [ dv*dp, dv*dv ]
+])
+
+H = np.matrix([
+    [ 1., 0. ],
+])
+
+I = mt.identity(M, dtype = np.float)
+
+def arrayOfMatrices(n, shape):
+    return np.asarray([mt.zeros(shape, dtype = np.float) for i in xrange(n)])
+
+# Base on the scipy-cookbook http://scipy-cookbook.readthedocs.io/items/KalmanFiltering.html
+sz_state = (M, 1)
+sz_cov   = (M, M)
+sz_K     = (M, 1)
+xhat= arrayOfMatrices(N, sz_state)      # a posteri estimate of x
+P=arrayOfMatrices(N, sz_cov)         # a posteri error estimate
+xhatminus=arrayOfMatrices(N, sz_state) # a priori estimate of x
+Pminus=arrayOfMatrices(N, sz_cov)    # a priori error estimate
+K=arrayOfMatrices(N, sz_K)         # gain or blending factor
+
+P[0] = mt.ones((M,M)) * 100
+xhat[0] = measurement[0]
+
+for k in range(1,N):
+    # time update
+    xhatminus[k] = A * xhat[k-1]
+    Pminus[k]    = A * P[k-1] + Q
+
+    # measurement update
+    K[k] = Pminus[k] * H.T * np.linalg.inv( H * Pminus[k] * H.T + R )
+    xhat[k] = xhatminus[k] + K[k] * (measurement[k] - H * xhatminus[k])
+    P[k] = ( I - K[k] ) * Pminus[k]
+    
+t = np.arange(N) * dt
+plt.figure()
+plt.subplot(2,1,1)
+plt.plot(t, measurement,'k+',label='noisy measurements')
+plt.plot(t, xhat[:,0,0],'b-',label='position estimate')
+plt.plot(t, x, 'r-', label='ground truth')
+
+plt.subplot(2,1,2)
+plt.plot(t, xhat[:,1,0],'g-',label='velocity estimate')
+
+plt.show()
\ No newline at end of file
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