无迹卡尔曼Python代码

无迹卡尔曼滤波（Unscented Kalman Filter，UKF）是卡尔曼滤波的一种扩展，通过引入无迹变换，解决了在高斯分布非线性变换后不再是高斯分布的问题。Python实现无迹卡尔曼滤波的代码如下：

import numpy as np

# UKF参数设置
alpha = 0.001
beta = 2
kappa = 0

# 状态维度
n = 3

# 测量维度
m = 1

# 状态转移函数
def f(x):
    return np.array([[x+x], [x+x], [x]])

# 测量函数
def h(x):
    return x

# 状态转移矩阵
def F(x, dt):
    F = np.array([[1, dt, 0], [0, 1, dt], [0, 0, 1]])
    return F

# 测量矩阵
def H(x):
    return np.array([[1, 0, 0]])

# 过程噪声协方差矩阵
def Q(x, dt):
    q = 0.01
    Q = np.array([[q*dt**3/3, q*dt**2/2, 0], [q*dt**2/2, q*dt, 0], [0, 0, 0]])
    return Q

# 测量噪声协方差矩阵
def R(x):
    r = 0.1
    R = np.array([[r]])
    return R

# 初始状态和协方差矩阵
x = np.array([, , ])
P = np.array([[1, 0, 0], [0, 1, 0], [0, 0, 1]])

# UKF权重计算
lambda_ = alpha**2*(n+kappa)-n
c = n+lambda_
Wm = np.zeros(2*n+1)
Wc = np.zeros(2*n+1)
Wm = lambda_/c
Wc = lambda_/(c+(1-alpha**2+beta))
for i in range(1, 2*n+1):
    Wm[i] = Wc[i] = 1/(2*c)

# UKF主函数
def UKF(x, P, z, dt):
    # Sigma点计算
    X = np.zeros((n, 2*n+1))
    X[:,0] = x[:,0]
    A = np.linalg.cholesky((n+kappa)*P)
    for i in range(n):
        X[:,i+1]   = x[:,0]+A[:,i]
        X[:,i+n+1] = x[:,0]-A[:,i]

    # 预测均值和协方差矩阵
    x_ = np.zeros((n,1))
    for i in range(2*n+1):
        x_ += Wm[i]*f(X[:,i:i+1])
    P_ = Q(X[:,0:1], dt)
    for i in range(1, 2*n+1):
        P_ += Wc[i]*(f(X[:,i:i+1])-x_).dot((f(X[:,i:i+1])-x_).T)
    
    # 计算测量值和协方差矩阵
    Z = np.zeros((m, 2*n+1))
    for i in range(2*n+1):
        Z[:,i:i+1] = h(X[:,i:i+1])
    z_ = np.zeros((m,1))
    for i in range(2*n+1):
        z_ += Wm[i]*Z[:,i:i+1]
    S = R(z) + np.dot(np.dot(H(x_), P_), H(x_).T)
    for i in range(1, 2*n+1):
        S += Wc[i]*(Z[:,i:i+1]-z_).dot((Z[:,i:i+1]-z_).T)
    
    # 计算卡尔曼增益
    K = np.dot(np.dot(P_, H(x_).T), np.linalg.inv(S))
    
    # 更新状态和协方差矩阵
    x = x_ + np.dot(K, z-z_)
    P = P_ - np.dot(np.dot(K, S), K.T)
    
    return x,P

# 测试数据
T = 50
dt = 0.01
t = np.arange(0,T,dt)
N = len(t)
z = np.sin(t)

# UKF滤波器迭代计算
for i in range(N):
    x,P = UKF(x,P,z[i:i+1],dt)