**5. Experimental results**

Before applying the proposed ACO to grid map merging, the spectra-based map merging (SMM) [13] algorithm was applied to find a coarse MTM. The SMM is a well-known indirect grid map merging algorithm which extracts spectral information from grid maps by the Hough transform and finds an MTM by matching the spectral information based on the cross-correlations. The individual grid maps in a multi-robot system were as shown in **Figure 6**. To reduce the computation time, each grid map was represented by a binary image with occupied (white) and unoccupied (black) grids.

Firstly, the rotation angle was coarsely estimated by the SMM. The Hough spectra and the cross-correlation between them are shown in **Figure 7**. The SMM estimates the rotation angle by taking the angle corresponding the maximum crosscorrelation value. After rotating one of the individual grid maps by the estimated rotation angle, the SMM estimates the *x* and *y* translation amounts by taking the amounts corresponding the maximum *x* and *y* cross-correlation value. The *x* spectra and the *x* cross-correlations between them are shown in the top of **Figure 8**. Similarly, the *y* spectra and the *y* cross-correlations between them are shown in the bottom of **Figure 8**. The merged map by the rotation angle and the translation amounts estimated by the SMM is shown in **Figure 9**. The two individual grid maps were properly merged. But, they needs to be merged more accurately.

The proposed ACO for grid map merging was implemented based on an open source [14]. The settings for the ACO for grid map merging were as follow. The number of iterations was set to 50. The number of samples was set to 30. The number of ants *Nant* was set to 100. The graphical results of the ACO for grid map merging are shown in **Figure 10**, which indicates that the pheromones were properly updated as time goes and found the optimal configuration of *x* and *y* translation amounts. In other word, the proposed method was successfully conducted and found the best *x* and *y* translation amounts. By the best *x* and *y* translation amounts and the rotation angle estimated by the SMM, the two individual grid maps were

**Figure 6.** *Individual grid maps in a multi-robot system. (a) Individual grid map 1, M*<sup>1</sup> *(b) Individual grid map 2, M*2*:*

**Figure 7.** *Rotation angle estimation by the SMM.*

**Figure 8.** *Translation amounts estimation by the SMM.*

merged more accurately as shown in **Figure 11**. Comparing with **Figure 9**, we can say that the error in the merged grid map was reduced.

The quantitative evaluation of the accuracy of grid map merging can be conducted with the following measure:

$$\text{Accuracy index} = \frac{\sum\_{\mathbf{x}=\hat{a}\_1}^{\hat{a}\_2} \sum\_{\mathbf{y}=\hat{b}\_1}^{\hat{b}\_2} \mathbf{M}\_1(\mathbf{x}, \mathbf{y}) \cdot \hat{\mathbf{M}}\_2(\mathbf{x}, \mathbf{y})}{N\_{overlap}} \tag{7}$$

where *Noverlap* is the number of commonly occupied grids in the overlapped areas when two individual grid maps are maximally overlapped, which is a global true

*Grid Map Merging with Ant Colony Optimization for Multi-Robot Systems DOI: http://dx.doi.org/10.5772/intechopen.98223*

### **Figure 9.**

*The merged map by the SMM. The map 2 (green) was transformed by the SMM, and the transformed map 2 (red) was properly merged into map 1 (blue). However, they need to be merged more accurately.*

### **Figure 10.**

*ACO results for grid map merging. The red circles represent states in* x *and* y *areas. The left image represents the whole tours at each iteration. The middle image represents the best tour (the queen). The right image represents the pheromones along the tours.*

value and not given to robots. *M*^ <sup>2</sup> is the transformed *M*<sup>2</sup> by the ACO. *a*^<sup>1</sup> ≤*x*≤*a*^<sup>2</sup> and ^ *b*<sup>1</sup> ≤*y*≤ ^ *b*<sup>2</sup> are the whole ranges of the *x* and *y* coordinates of *M*<sup>1</sup> and *M*^ 2.

The map merging results of the proposed grid map merging method which uses both the SMM and the ACO was quantitatively compared with those of the only SMM-based grid map merging as shown in **Figure 12**. Because the performance of the ACO depends on the number of ants *Nant*, the accuracy indices of the proposed method were analyzed with various *Nant*. As expected, the ACO improved the accuracy of grid map merging with the SMM.

### **Figure 11.**

*The updated merged map by the ACO. The two individual maps were merged more accurately.*

### **Figure 12.**

*The improved accuracy of the grid map merging with the ACO.*

Although the accuracy index of the proposed method increases according to *Nant*, the differences were not significant.
