## 2 Improved AMCL Algorithm
### Improved Resampling to Avoid Kidnapping  
#### 2.4.1 Adjustment of KLD Dynamic Resampling  
The kidnapping problem refers to the fact that when the robot undergoes a drastic change in the environment or is moved to a new location, the particles in the particle filter will gather at the wrong location, resulting in inaccurate localization. In order to effectively reduce or avoid the kidnapping problem in the AMCL algorithm, this can be achieved by improving the resampling strategy of the particle filter. Traditional resampling methods, such as uniform resampling, are prone to lead to the kidnapping problem. Therefore, the kidnapping problem is avoided by using an improved KLD (Kullback-Leibler divergence resampling) sampling method to decide whether to resample or not by calculating the information entropy of the particle weights. The Kullback-Leibler distance represents the approximation error between two probability distributions \(p\) and \(q\), i.e.,  
\[K(p,q)=\sum_{x}p(x)log\frac{p(x)}{q(x)} \tag{7}\]  
The Kullback-Leibler distance is non-negative and has a value of zero if and only if the two probability distributions \(p\) and \(q\) agree. Assuming there is a discrete distribution with \(k\) different subspaces, where the vector \(X=\{X_{1},X_{2},X_{3},\ldots,X_{k}\}\) represents the number of particles sampled from each subspace and the vector \(P=\{P_{1},P_{2},P_{3},\ldots,P_{k}\}\) represents the true probability of each subspace, the maximum likelihood estimation probability density is \(\hat{P}=\frac{X}{n}\). When n satisfies a certain number, it can be ensured that the Kullback-Leibler distance \(K\left(\hat{P},P\right)\) between the true probability density and the estimated probability density is less than a threshold value \(\epsilon\), which ensures that the approximation error between the true probability density and the estimated probability density is minimized. At this point, according to the Wilson Ferty transformation method, the approximate calculation formula for \(n\) with the minimum approximation error can be obtained as follows:  
\[n=\frac{k-1}{2\epsilon}\{1-\frac{2}{9(k-1)}+\sqrt{\frac{2}{9(k-1)}}z_{1-\delta }\}^{3} \tag{8}\]  
where \(z_{1-\delta}\) is the standard normal distribution for the upper quartile \(1-\delta\). From the above equation, the number of particles required to minimize the approximation error between the estimated and true posterior distributions can be obtained. The main purpose of particle filtering is to estimate a posterior distribution, so it is only necessary to determine the number of effective subspaces \(k\) and the pre-given \(\epsilon\) and \(\delta\) to obtain the minimum number of particles required to ensure the estimation performance of particle filtering. It follows that the the improved KLD dynamically adjusted resampling process is as follows:  
1. Input the collection of particles \(S_{t-1}=\left\{\left(x_{t-1}^{(i)},w_{t-1}^{(i)}\right)\middle|i=1,\ldots,n\right\}\) after resampling at moment \(t-1\), the observation B, set \(\epsilon\) and \(\delta\), the minimum value of the total number of particles \(n_{xmin}\), and the number of particles \(n_{t-1}\) at moment \(t-1\). 2. Set the predicted particle set at time \(t\) to \(\overset{\sim}{s}_{t}=\varnothing\), with a total number of particles of \(n=1500\), a number of particles on the line \(n_{xmax}=100000\), and variables \(k\) and \(\alpha\)both being 0. Use the dynamic adjustment method to resample particle \(x_{t-1}^{(i)}\) to obtain the predicted particle \(x_{t}^{(i)}\), and calculate the corresponding weight \(w_{t}^{(i)}\). 3. Accumulation of weights: \(\widetilde{s}_{t}=\widetilde{s}_{t}\cup\left\{x_{t}^{(n)},w_{t}^{(n)}\right\}\), place the newly predicted particles into the set of predicted particles \(b\). If \(x_{t}^{(n)}\) falls into the space interval \(b\), then \(k=k+1\), while the interval \(b\) becomes non-empty. 4.