The estimation may have one or more peak values, which are supposed to be as relative selleck inhibitor pose hypotheses between m1 and m2. Suppose that homogeneous transformation matrix T is one of these hypotheses, as show in Figure 5, the homogeneous transformation matrix T3 of the two robots’ relative pose can be calculated with T, T1, and T2 as follows:T3=T1?1?T?T2.(10)Figure 5Relative pose of robots.We can use T3 to path planning two robots’ paths in the map m1 and make them meet as far as possible. The plan is operated on Robot1, and the results are sent to Robot2 in order to collaborate with each other. If the two robots can meet according to the path planning, the relative pose hypothesis T is true, and we go to the stage of estimation optimization; otherwise we test the next hypothesis.

If all hypotheses are false, the robot still constructs map independently (single-robot SLAM), waiting for the next operation of map merging.5. Estimation OptimizationAfter the previous steps, the preliminary estimation to the relative pose of map is got. Based on the research of literature [24], the relative pose is further optimized. The concrete optimization process is as follows.5.1. Dissimilarity Measurement Function of MapFrom the definition of map merging, the map merging problem can be seen as an optimization problem, whose optimization function is ��(). It is the problem that ��() is directly used as optimization function. Because the values of ��(m1, T(tx,ty,t��)(m2)) are arbitrarily leap with continuous changes of variables (tx, ty, t��), the function ��() delivers no effective gradients to do optimization like hill-climbing algorithm.

The dissimilarity function is as follows:��(m1,m2)=��c��C[d(m1,m2,c)+d(m1,m2,c)](11)withd(m1,m2,c)=��m1[p1]=cmin??md(p1,p2)?�O?m2[p2]=c#c(m1),(12)where C denotes grid range of map m1 and map m2, m1[p1] denotes the value of grid p1 in map m1, md(p1, p2) = |x1 ? x2| + |y1 ? y2| denotes the Manhattan-distance between points p1 and p2, and #c(m1) = # m1[p1] = c denotes the number of grids with value c in map m1.To simplify the calculation, the grids’ value of the map is marked as ��free�� or ��occupied�� or ��unknown�� according to the predefined threshold. Only occupied and free grids are considered for computing dissimilarity function, so C = occ, free. In order to compute the dissimilarity function in linear time, a so called distance-map is introduced.

Distance-map dmapc[x1][y1] denotes the Manhattan distance between the grid p1 = (x1, y1) in map m1 and a grid which GSK-3 is the nearest point to p1 with value c in map m2:dmapc[x1][y1]=min?m2[p2]=c.(13)The concrete calculation process for dmapc is shown in Algorithm 1.Algorithm 1Calculation process of distance-map dmapc in grid map.Using the distance-map dmapc, we can calculate d(m1, m2, c) with Algorithm 2.Algorithm 2Calculation process of d(m1, m2, c).5.2.