As a primary stage, an abstraction procedure is needed that should allow both the robots’ environment modelling and the obtaining of the necessary data for the planning and control mechanisms. It will involve the partitioning of the environment into a set of adjacent cells having the same shape. Such partitions (also called cell decompositions) can be created by using tools from computational geometry (Berg et al., 2008), the name of the partition being given by the shape of its cells (Choset et al., 2005, La Valle, 2006). Once a partition is created, each cell corresponds to a node of a graph in the abstract representation. The edges between nodes correspond to adjacency relations between cells and to control capabilities of robot for moving from one cell to an adjacent one. Then, for designing these control laws, one can use results from (Habets and van Schuppen, 2004), where feedback control laws driving all trajectories of an affine system from a polytopal or simpliceal region through a desired facet were designed. Thus, the motion and control capabilities of robots are abstracted into a finite graph, where a node (place) corresponds to a region where the robot can be located, and an edge (way) corresponds to a feedback control law driving the robot from one place to another. For a better Fig. 3. A multi-agent architecture to solve the navigation for two mobile robots Fig. 5. The environment map and initial positions of robots Fig. 6. The environment with several obstructed ways correspond to the ways between positions 5 and 8, 24 and 32 which have the weight of /5 . The optimal path for this map consists in the following succession of positions: 1 - 2 - 4 - 11- 17- 21- 27- 33 - 36 - 38. In fact, the Agent 1 (the one for the red robot) detected the path from position 1 to 21, and the Agent 2 the path from 38 to 11, and then the node corresponding to the position labelled 21 is detected by the Agent 1 in both agents’ paths and the searching processes were ended, the result being the optimal solution. The image in Fig. 6 shows that in our experiment five ways were successively detected as being obstructed. Fig. 7. The robots in the meeting position Fig. 8. A case with a large number of blocked ways
