组合导航知识回顾（1）

参考博客：无人驾驶汽车系统入门（一）——卡尔曼滤波与目标追踪_AdamShan的博客-CSDN博客_卡尔曼滤波预测汽车轨迹

import numpy as np
%matplotlib inline
import matplotlib.pyplot as plt
from scipy.stats import norm
from sympy import Symbol, Matrix                     #科学计算库，可计算方程、微风、无穷级数
from sympy.interactive import printing               #可打印出根号、求集合、无穷级数——为数学表达式打印unicode字符
printing.init_printing()



## Initialize related variables

x = np.matrix([[0.0,0.0,0.0,0.0]]).T              # 初始状态
P = np.diag([1000.0,1000.0,1000.0,1000.0])        # 提前预测误差协方差

dt = 0.1                                          # 时间片
F = np.matrix([[1.0,0.0,dt,0.0],                  # 预测矩阵F(4,4)
               [0.0,1.0,0.0,dt],
               [0.0,0.0,1.0,0.0],
               [0.0,0.0,0.0,1.0]])

H = np.matrix([[0.0,0.0,1.0,0.0],                #  测量矩阵H(2,4)
               [0.0,0.0,0.0,1.0]])

Ra = 10.0**2
R = np.matrix([[Ra,0.0]                         # 测量噪声协方差矩阵R
               ,[0.0,Ra]])

pv = 0.5
G = np.matrix([[0.5*dt**2],
              [0.5*dt**2],
              [dt],
              [dt]])
Q = G*G.T*pv**2                                # 过程噪声的协方差矩阵Q

I = np.eye(4)                                  #  单位矩阵



## Initialize sensor data and graphs

m = 200        # Measurements
vx= 20         # in X
vy= 10         # in Y

mx = np.array(vx+np.random.randn(m))
my = np.array(vy+np.random.randn(m))
measurements = np.vstack((mx,my))                  #  表示传感器的测量数据
#print('Standard Deviation of Velocity Measurements=%.2f' % np.std(mx))

fig = plt.figure(figsize=(16,5))
plt.step(range(m),mx, label='$dot x$')
plt.step(range(m),my, label='$dot y$')
plt.ylabel(r'Velocity $m/s$')
plt.title('Measurements')
plt.legend(loc='best',prop={'size':18})



## Data in the stored procedure 
xt = []
yt = []
dxt= []
dyt= []
Zx = []
Zy = []
Px = []
Py = []
Pdx= []
Pdy= []
Rdx= []
Rdy= []
Kx = []
Ky = []
Kdx= []
Kdy= []

def savestates(x, Z, P, R, K):    # x--状态   Z--观测值  P--提前预测误差协方差  R--测量噪声协方差矩阵  K--卡尔曼增益
    xt.append(float(x[0]))
    yt.append(float(x[1]))
    dxt.append(float(x[2]))
    dyt.append(float(x[3]))
    Zx.append(float(Z[0]))
    Zy.append(float(Z[1]))
    Px.append(float(P[0,0]))
    Py.append(float(P[1,1]))
    Pdx.append(float(P[2,2]))
    Pdy.append(float(P[3,3]))
    Rdx.append(float(R[0,0]))
    Rdy.append(float(R[1,1]))
    Kx.append(float(K[0,0]))
    Ky.append(float(K[1,0]))
    Kdx.append(float(K[2,0]))
    Kdy.append(float(K[3,0]))    

    

## KF function    
for n in range(len(measurements[0])):
 
    # Time Update (Prediction)
    # ========================
    # Project the state ahead
    x = F*x
    
    # Project the error covariance ahead
    P = F*P*F.T + Q
    
    # Measurement Update (Correction)
    # ===============================
    # Compute the Kalman Gain
    S = H*P*H.T + R
    K = (P*H.T) * np.linalg.pinv(S)
    
    # Update the estimate via z
    Z = measurements[:,n].reshape(2,1)
    y = Z - (H*x)                            # Innovation or Residual
    x = x + (K*y)
    
    # Update the error covariance
    P = (I - (K*H))*P
    
    # Save states (for Plotting)
    savestates(x, Z, P, R, K)


## figure
def plot_velocity():
    fig = plt.figure(figsize=(16,9))
    plt.step(range(len(measurements[0])),dxt, label='$estimateVx$')
    plt.step(range(len(measurements[0])),dyt, label='$estimateVy$')
    
    plt.step(range(len(measurements[0])),measurements[0], label='$measurementVx$')
    plt.step(range(len(measurements[0])),measurements[1], label='$measurementVy$')

    plt.axhline(vx, color='#999999', label='$trueVx$')
    plt.axhline(vy, color='#999999', label='$trueVy$')

    plt.xlabel('Filter Step')
    plt.title('Estimate (Elements from State Vector $x$)')
    plt.legend(loc='best',prop={'size':11})
    plt.ylim([0, 30])
    plt.ylabel('Velocity')


def plot_path():
    fig = plt.figure(figsize=(16,16))
    plt.scatter(xt,yt, s=20, label='State', c='k')
    plt.scatter(xt[0],yt[0], s=100, label='Start', c='g')
    plt.scatter(xt[-1],yt[-1], s=100, label='Goal', c='r')

    plt.xlabel('X')
    plt.ylabel('Y')
    plt.title('Position')
    plt.legend(loc='best')
    plt.axis('equal')
    
plot_velocity()
plot_path()