Kalman理论介绍

一. 简单理论介绍理论

二. 升华理论介绍

Kalman基本应用

一. Kalman跟踪/滤波

对单个数据滤波，无法建立运动学模型

通过建立和自身相关的状态方程即可

是一种平滑操作（上一时刻和当前时刻的关系）

举例：

对一个平面运动的质点进行跟踪（ $X、Y$ ）？

速度 $v、\alpha，\omega$ 都是未知状态

求解：

fig = plt.figure()
axis = fig.add_subplot(1,1,1)

func_data = lambda x : x + x^2
z = np.mat(func_data(np.arange(1,100)))
x_mat = np.mat([[0,],[0.]])#状态矩阵[x,delta_x]
p_mat = np.mat([[1, 0], [0, 1]])#状态协方差矩阵
f_mat = np.mat([[1, 1],[0.,1.]])#状态转移矩阵
q_mat = np.mat([[0.0001, 0], [0, 0.0001]])
h_mat = np.mat([1.,0])# 观测矩阵[x]
r_mat = np.mat([1])#观测协方差矩阵
result = []
for i in range(z.shape[1]):
    x_predict = f_mat * x_mat
    p_predict = f_mat * p_mat * f_mat.T + q_mat
   
    kalman = p_predict * h_mat.T / (h_mat * p_predict * h_mat.T + r_mat)  
    
    x_mat = x_predict + kalman *(z[0, i] - h_mat * x_predict)
    p_mat = (np.eye(2) - kalman * h_mat) * p_predict
    result.append(x_predict[0,0])

axis.plot(result,label='predict')
axis.plot(z.tolist()[0],label='groundtruth')
axis.legend()

image

二. Kalman预测/融合（单传感器）

[x] 运动学模型
[x] 单一传感器
[x] 速度 $v、\alpha，\omega$ 推导可知

举例一：

问题和例子文档

一个运动小车的位置和速度的测量等信息可以被测量(一个传感器)，也可以通过牛顿运动学方程进行解算，这两个到底谁占的比例高？使用Kalman的协方差矩阵进行比例的计算。。。。具体看文档

image

举例二：

stackoverflow上的一个例子

import numpy as np
import matplotlib.pyplot as plt

def kalman_xy(x, P, measurement, R,
              motion = np.matrix('0. 0. 0. 0.').T,
              Q = np.matrix(np.eye(4))):
    """
    Parameters:    
    x: initial state 4-tuple of location and velocity: (x0, x1, x0_dot, x1_dot)
    P: initial uncertainty convariance matrix
    measurement: observed position
    R: measurement noise 
    motion: external motion added to state vector x
    Q: motion noise (same shape as P)
    """
    return kalman(x, P, measurement, R, motion, Q,
                  F = np.matrix('''
                      1. 0. 1. 0.;
                      0. 1. 0. 1.;
                      0. 0. 1. 0.;
                      0. 0. 0. 1.
                      '''),
                  H = np.matrix('''
                      1. 0. 0. 0.;
                      0. 1. 0. 0.'''))

def kalman(x, P, measurement, R, motion, Q, F, H):
    '''
    Parameters:
    x: initial state
    P: initial uncertainty convariance matrix
    measurement: observed position (same shape as H*x)
    R: measurement noise (same shape as H)
    motion: external motion added to state vector x
    Q: motion noise (same shape as P)
    F: next state function: x_prime = F*x
    H: measurement function: position = H*x

    Return: the updated and predicted new values for (x, P)

    See also http://en.wikipedia.org/wiki/Kalman_filter

    This version of kalman can be applied to many different situations by
    appropriately defining F and H 
    '''
    # UPDATE x, P based on measurement m    
    # distance between measured and current position-belief
    y = np.matrix(measurement).T - H * x
    S = H * P * H.T + R  # residual convariance
    K = P * H.T * S.I    # Kalman gain
    x = x + K*y
    I = np.matrix(np.eye(F.shape[0])) # identity matrix
    P = (I - K*H)*P

    # PREDICT x, P based on motion
    x = F*x + motion
    P = F*P*F.T + Q

    return x, P

def demo_kalman_xy():
    x = np.matrix('0. 0. 0. 0.').T 
    P = np.matrix(np.eye(4))*1000 # initial uncertainty

    N = 20
    true_x = np.linspace(0.0, 10.0, N)
    true_y = true_x**2
    observed_x = true_x + 0.05*np.random.random(N)*true_x
    observed_y = true_y + 0.05*np.random.random(N)*true_y
    plt.plot(observed_x, observed_y, 'ro')
    result = []
    R = 0.01**2
    for meas in zip(observed_x, observed_y):
        x, P = kalman_xy(x, P, meas, R)
        result.append((x[:2]).tolist())
    kalman_x, kalman_y = zip(*result)
    plt.plot(kalman_x, kalman_y, 'g-')
    plt.show()

demo_kalman_xy()

这部分比较简单，网上的例子大部分都是基于此的。。。