Kalman Filter (python)

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对卡尔曼滤波算法的一些理解和思考，以python形式呈现

 

// 取位置初值为0，速度初值为10m/s
// T为采样周期，q为过程噪声方差，r为量测噪声方差
// N为蒙特卡洛模拟次数
// 初始状态协方差矩阵 \[P=\left[\begin{array}{cc}R & R / T \\ R / T & 2 R / T^{2}\end{array}\right]\]

源代码：

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import numpy as np
import matplotlib.pyplot as plt

T = 1
q = 1
r = 5
R = r
Q = q
N = 50      # Monte Carlo simulation times
 
M_measPos = []
M_estPos = []
M_estVel = []
M_measTime = []
M_estDifPos = []
M_estDifVel = []
x_rme = []
x_rmse = []
x_rrmse = []
v_rme = []
v_rmse = []
v_rrmse = []

for i in range(N):
    def getMeasurement(updateNumber):
        if updateNumber == 1:
          getMeasurement.currentPosition = 0 # initial position
          getMeasurement.currentVelocity = 10 # m/s  initial velocity

        w = np.random.normal(loc=0,scale=np.sqrt(r),size=1)
        v = np.random.normal(loc=0,scale=np.sqrt(q),size=1)

        x = getMeasurement.currentPosition + getMeasurement.currentVelocity*T + v
        getMeasurement.currentPosition = x - v
        getMeasurement.currentVelocity = 10 + w
        return [x, getMeasurement.currentPosition, getMeasurement.currentVelocity] 

    def filter(x, updateNumber):
        # Initialize State
        if updateNumber == 1:
            filter.X = np.array([[0],
                                [10]])
            filter.P = np.array([[R, R/T],
                                [R/T, 2*R/(T**2)]])
            filter.F = np.array([[1, T],
                                [0, 1]])
            filter.H = np.array([[1, 0]])
            filter.HT = np.array([[1],
                                 [0]])
            filter.Gamma = np.array([[T**2/2],
                                    [T]])  
            filter.R = R
            filter.Q = Q
           
        # Predict State Forward
        X_prime = filter.F.dot(filter.X)
        # Predict Covariance Forward
        P_prime = filter.F.dot(filter.P).dot(filter.F.T) + filter.Gamma.dot(filter.Q).dot(filter.Gamma.T)
        # Compute Kalman Gain
        S = filter.H.dot(P_prime).dot(filter.HT) + filter.R
        K = P_prime.dot(filter.HT)*(1/S)
        # Estimate State
        residual = x - filter.H.dot(X_prime)
        filter.X = X_prime + K*residual
        # Estimate Covariance
        filter.P = P_prime - K.dot(filter.H).dot(P_prime)
        return [filter.X[0], filter.X[1], filter.P]

    def testFilter():
        t = np.linspace(0, 1, num=100)
        numOfMeasurements = len(t)

        measTime = []
        measPos = []
        measDifPos = []
        estDifPos = []
        estDifVel = []
        estPos = []
        estVel = []
        x_rme = []
        x_rmse = []
        x_rrmse = []
        v_rme = []
        v_rmse = []
        v_rrmse = []

        for k in range(1,numOfMeasurements):
            x = getMeasurement(k)
            # Call Filter and return new State
            f = filter(x[0], k)
            # Save off that state so that it could be plotted
            measTime.append(k)
            measPos.append(x[0])
            measDifPos.append(x[0]-x[1])
            estDifPos.append(f[0]-x[1])
            x_rme.append(np.abs((f[0]-x[1])/x[1]))
            x_rmse.append((f[0]-x[1])**2)
            x_rrmse.append(((f[0]-x[1])/x[1])**2)
            estDifVel.append(f[1]-x[2])
            v_rme.append(np.abs((f[1]-x[2])/x[2]))
            v_rmse.append((f[1]-x[2])**2)
            v_rrmse.append(((f[1]-x[2])/x[2])**2)
            estPos.append(f[0])
            estVel.append(f[1])
            posVar = f[2]
        
        return [measTime, measPos, estPos, estVel, estDifPos, estDifVel, 
               x_rme, v_rme, x_rmse, v_rmse, x_rrmse, v_rrmse]

    t = testFilter()

    M_measTime.append(t[0])
    M_measPos.append(t[1])  
    M_estPos.append(t[2])
    M_estVel.append(t[3])
    M_estDifPos.append(np.abs(t[4]))
    M_estDifVel.append(np.abs(t[5]))
    x_rme.append(t[6])
    v_rme.append(t[7])
    x_rmse.append(t[8])
    v_rmse.append(t[9])
    x_rrmse.append(t[10])
    v_rrmse.append(t[11])

x_ME = np.mean(M_estDifPos, axis=0)             # mean error of position
print(np.mean(x_ME))
x_RME = np.mean(x_rme, axis=0)                  # relative mean error of position
print(np.mean(x_RME))
x_RMSE = np.sqrt(np.mean(x_rmse, axis=0))       # root mean square error of position
print(np.mean(x_RMSE))
x_RRMSE = np.sqrt(np.mean(x_rrmse, axis=0))     # relative root mean square error of position
print(np.mean(x_RRMSE))

v_ME = np.mean(M_estDifVel, axis=0)             # mean error of velocity
print(np.mean(v_ME))
v_RME = np.mean(v_rme, axis=0)                  # relative mean error of velocity
print(np.mean(v_RME))
v_RMSE = np.sqrt(np.mean(v_rmse, axis=0))       # root mean square error of velocity
print(np.mean(v_RMSE))
v_RRMSE = np.sqrt(np.mean(v_rrmse, axis=0))     # relative root mean square error of velocity
print(np.mean(v_RRMSE))

# ##################### plot ######################
# ###### one simulation results #####
# plot1 = plt.figure(1)
# plt.plot(t[0], t[2])
# plt.ylabel('Position')
# plt.xlabel('Time')
# plt.title('Position Estimate On one Measurement Update \n', fontweight="bold")
# plt.legend(['Estimate Position'])
# plt.grid(True)

# plot2 = plt.figure(2)
# plt.plot(t[0], t[3])
# plt.ylabel('Velocity (meters/seconds)')
# plt.xlabel('Update Number')
# plt.title('Velocity Estimate On one Measurement Update \n', fontweight="bold")
# plt.legend(['Estimate Velocity'])
# plt.grid(True)

# ###### error analysis with Monte Carlo simulation #####
# plot3 = plt.figure(3)
# plt.plot(t[0], x_ME)
# plt.title('Position Mean Errors \n', fontweight="bold")
# plt.ylabel('ME (meters)')
# plt.xlabel('Update Number')
# plt.grid(True)
# plt.xlim([0, 100])
# plt.show()

# plot4 = plt.figure(4)
# plt.plot(t[0], x_RME)
# plt.title('Position Relative Mean Errors \n', fontweight="bold")
# plt.ylabel('RME')
# plt.xlabel('Update Number')
# plt.grid(True)
# plt.xlim([0, 100])
# plt.show()

# plot5 = plt.figure(5)
# plt.plot(t[0], x_RMSE)
# plt.title('Position Root Mean Square Errors \n', fontweight="bold")
# plt.ylabel('RMSE')
# plt.xlabel('Update Number')
# plt.grid(True)
# plt.xlim([0, 100])
# plt.show()

# plot6 = plt.figure(6)
# plt.plot(t[0], x_RRMSE)
# plt.title('Position Relative Root Mean Square Errors  \n', fontweight="bold")
# plt.ylabel('RRMSE')
# plt.xlabel('Update Number')
# plt.grid(True)
# plt.xlim([0, 100])
# plt.show()


# plot7 = plt.figure(7)
# plt.plot(t[0], v_ME)
# plt.title('Velocity Mean Errors \n', fontweight="bold")
# plt.ylabel('ME (meters/s)')
# plt.xlabel('Update Number')
# plt.grid(True)
# plt.xlim([0, 100])
# plt.show()

# plot8 = plt.figure(8)
# plt.plot(t[0], v_RME)
# plt.title('Velocity Relative Mean Errors \n', fontweight="bold")
# plt.ylabel('RME')
# plt.xlabel('Update Number')
# plt.grid(True)
# plt.xlim([0, 100])
# plt.show()

# plot9 = plt.figure(9)
# plt.plot(t[0], v_RMSE)
# plt.title('Velocity Root Mean Square Errors \n', fontweight="bold")
# plt.ylabel('RMSE')
# plt.xlabel('Update Number')
# plt.grid(True)
# plt.xlim([0, 100])
# plt.show()

# plot10 = plt.figure(10)
# plt.plot(t[0], v_RRMSE)
# plt.title('Velocity Relative Root Mean Square Errors  \n', fontweight="bold")
# plt.ylabel('RRMSE')
# plt.xlabel('Update Number')
# plt.grid(True)
# plt.xlim([0, 100])
# plt.show()

#代码部分修改自网络#