Merge pull request AtsushiSakai#205 from rsasaki0109/add_localization_with_ensemble_kalman_filter

add localization with ensemble kalman filter

Author: AtsushiSakai

Localization/ensemble_kalman_filter/ensemble_kalman_filter.py

@@ -0,0 +1,232 @@
+"""
+
+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, 4))
+
+    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)
+        angle = pi_2_pi(math.atan2(dy, dx) - xTrue[2, 0])
+        if d <= MAX_RANGE:
+            dn = d + np.random.randn() * Qsim[0, 0]  # add noise
+            anglen = angle + np.random.randn() * Qsim[1, 1]  # add noise
+            zi = np.array([dn, anglen,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_LM_Pos(x, landmarks):
+    landmarks_pos=np.zeros((2*landmarks.shape[0],1))
+    for (i,lm) in enumerate(landmarks):
+        landmarks_pos[2*i] = x[0, 0] + lm[0] * math.cos(x[2, 0] + lm[1]) + np.random.randn() * Qsim[0, 0]/np.sqrt(2)
+        landmarks_pos[2*i+1] = x[1, 0] + lm[0] * math.sin(x[2, 0] + lm[1]) + np.random.randn() * Qsim[0, 0]/np.sqrt(2)
+    return landmarks_pos
+
+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]*2, 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]
+        z_pos=calc_LM_Pos(x, z)
+        pz[:, ip] = z_pos[:, 0]
+
+    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
+
+    z_lm_pos = z[:,[2,3]].reshape(-1,)
+
+    px_hat=px + K @ (np.tile(z_lm_pos,(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 pi_2_pi(angle):
+    return (angle + math.pi) % (2 * math.pi) - math.pi
+
+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, 2]], [xTrue[1, 0], z[i, 3]], "-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()