一些简单地想法 - Something TO DO
一些简单地想法
Something TO DO
posted @ 2014年10月10日 16:05
in Multi-Agent Systems
with tags
Nearest neighbor approach
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简单的基础知识
Nearest neighbor approach
The nearest neighbor algorithm was one of the first algorithms used to determine a solution to the travelling salesman problem. In it, the salesman starts at a random city and repeatedly visits the nearest city until all have been visited. It quickly yields a short tour, but usually not the optimal one.
These are the steps of the algorithm:
- stand on an arbitrary vertex as current vertex.
- find out the shortest edge connecting current vertex and an unvisited vertex V.
- set current vertex to V.
- mark V as visited.
- if all the vertices in domain are visited, then terminate.
- Go to step 2.
但是此处我们不用最近邻算法，我们采用匈牙利算法来做匹配。
The Ieteration Closest Point Algorithm
写一个小程序作为例子来说明迭代最近点算法在多智能体系统编队中的应用。
六个智能体，可以组成的编队比较多样。组成TEAM队形
1，迭代最近点算法是用来寻找两个点集之间最佳匹配的算法。
2，问题描述：
·Given: 两个相应的点集：$X={x_1,\ldots,x_n}$,$P={p_1,\ldots,p_n}$
·Wanted:求解平移量$T$和旋转矩阵$R$，使得下式平方差的和最小：
\[E(R,T)=\frac{1}{N_p}\sum ^{N_p}_{i=1}||x_i-Rp_i-t||^2\]
其中，$x_i$和$p_i$是对应点集中的点。
3，求解方法：
·首先计算两个点集的几何中心：
\[u_x=\frac{1}{N_x}\sum_{i=1}^{N_x}x_i\]
\[u_p=\frac{1}{N_p}\sum_{i=1}^{N_p}p_i\]
·计算点集中每个点与几何中心的差值：
\[X'=\{x_i-u_x\}=\{x_i'\}\]
\[P'=\{p_i-u_p\}=\{p_i'\}\]
·做一个SVD分解：
令$W=\sum_{i=1}^{N_p}x_i'p_i'^T$,
其中，$U,V\in R^{3\times 3}$ 是正定的，并且$\sigma_1>=\sigma_2>=\sigma_3$是$W$的奇异值。
·后边不加证明的给出定理：
如果$rand(W)=3$，使得$E(R,T)$最优的$R$和$T$是惟一的：
\[R=UV^T\]
\[T=u_x-Ru_p\]
仿真参考：用matlab写的针对3D数据点的匹配仿真【1】。
根据以上仿真改写2D数据点的匹配仿真。
2·然后是实际仿真
应用匈牙利算法的迭代最近点算法
【1】参考文献：http://www.mathworks.com/matlabcentral/fileexchange/27804-iterative-closest-point
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