Kalman Filter is build according to the assumptions of f linear state transitions and linear measurements with added Gaussian noise,but which are rarely fulfilled in practice.The extended Kalman filter (EKF) overcomes one of these assumptions: the linearity assumption.Here the assumption is that the next state probability and the measurement probabilities are governed by nonlinear functions g and h,respectively:
xtzt=g(ut,xt−1)+εt=h(xt)+δt(1)
The function g replace the matrixs A and B ,and the h replace the matrix H.Unfortunately,the belief is no longer a Gaussian.In fact, performing the belief update exactly is usually impossible.
(写不下去了，切换中文，想想写paper多难。。。)
Linearization Via Taylor Expansion
EKF使用称为 Taylor Expansion 的方法来线性化非线性函数。泰勒展开就是在某点的值和导数构造一个线性approximation。斜率是对某点的偏导数：
g′(ut,xt−1):=∂xt−1∂g(ut,xt−1)
其中 g 的值和斜率是取决于 g 的参数的，选取的逻辑就是在线性化的时候最有可能的值，对高斯函数，最有可能的状态就是后验的均值 μt−1 。换句话说就是 g 在 μt−1 处 approximate：
μt equal to x^k ,represent update the state at k of estimation with measurement。
μt−1 equal to x^k−1 ,represent the state at k−1 of estimation。
Σˉt equal to Pk− , a priori estimate error covariance,说人话就是估计的误差协方差矩阵，利用先验的知识，还没有得到观测数据。
Σt−1 equal to Pk−1 ,a posteriori estimate error covariance,说人话就是利用后验估计误差协方差矩阵，就是利用获得的观测数据更新了误差协方差矩阵。
Gt ，At and A ,A present a constant transform matrix,but At present a dynamic, Gt is a Jacobian matrix,what is Jacobian matrix?Partial derivative of variable matrix.
Ht and H and Ct Jacobian Ht corresponds to Ct .