姓名：刘敏提     学号：20021110076     学院：电子工程学院

转自：微信公众号 “ 休语的复杂网络分析实验室”。

【嵌牛导读】贝叶斯递推状态估计算法。 想常用的卡尔曼滤波（KF），扩展卡尔曼滤波（EKF），无迹卡尔曼滤波（UKF），粒子滤波（PF）都是通过不同的假设来近似最优贝叶斯滤波得到的。在贝叶斯框架下，通过动态参数的先验概率密度和观测似然函数来求解感兴趣参数的后验概率密度。其在目标定位、跟踪中得到广泛应用。

【嵌牛鼻子】BF、KF，EKF，UKF 、PF

【嵌牛提问】这些滤波算法的推导过程是？

【嵌牛正文】

定位问题和卡尔曼滤波

定位问题一个最核心的特点就是存在这两种误差：运动误差和测量误差。我们考虑一个无人驾驶的小车。车辆通过自身的运动方式（比如通过速度和方向）推算出来的他可能的位置，这里存在的误差叫“运动误差”。此外，外界可以通过测量得到小车的位置，这里面存在的误差叫“测量误差”。

定位问题的各种滤波方法都是为了处理这两种误差。

卡尔曼滤波通常用于跟踪和定位加性高斯误差的线性系统。

卡尔曼滤波的公式及助记方式：

三种不确定度P_0，Q和R对于卡尔曼滤波的影响：

（1）P_0的影响：

P_0->适当的设置：x_0准确，P_0小：

• This looks fantastic at first blush. The plot does not have the spike that the former plot did; the filter starts tracking the measurements and doesn’t take any time to settle to the signal. 

• However, there is an initial spike for the variance in position, and that it never really converges. 

• Using a small P will avoid having the Kalman filter take time to accurately track the signal, but if we are truly uncertain about the initial measurements this can cause the filter to generate very bad results. 

• For your Kalman filter to perform well you must set P to a value that truly reflects your knowledge about the data.

P_0->适当的设置：x_0不准确，P_0大：

• The Kalman filter is very uncertain about the initial state, so it converges onto the signal much faster. It is producing good output after only 5 to 6 evolutions. 

• If we do not know where the object is when we start tracking, we would normally take the first measurement, use that to initialize the Kalman filter, and proceed from there.

P_0->不适当的设置：x_0准确，P_0大：

Looking at the output we see a very large spike in the filter output at the beginning. If you look at the data (dotted red line) you will see a corresponding, smaller spike in the beginning of the data. We set P=500. That is a pretty large error, so when the measurement spikes the Kalman filter distrusts its own estimate and jumps wildly to try to incorporate the measurement. Then, as the filter evolves P quickly converges to a more realistic value. 

P_0->不适当的设置：x_0不准确，P_0小：

• The initial estimates are terrible and that it takes the filter a long time to start converging onto the signal • This is because we told the Kalman filter that we strongly believe in our initial estimate of 100m and were incorrect in that belief.

（2）Q的影响：

Q->适当的设置：系统方程准确，Q小：

Q->不适当的设置：系统方程准确，Q大：

（3）R的影响：

R->适当的设置：测量值准确，R小：

R->不适当的设置：测量值准确（预测值不准确），R大：

扩展卡尔曼滤波（EKF）和无迹卡尔曼滤波（UKF）

扩展卡尔曼滤波的公式：

无迹卡尔曼滤波的公式：

粒子滤波 

首先要记住这个重要的公式：

然后是如下助记图示：

最后是对应的文字叙述：