In the next step a new fuzzy relation T is computed by using sub-triangle product <3 to fuzzy relation R and RT, the transposed relation of R. The fuzzy relation T as shown in (9) is the product relation between candidate set S that means the degree of implication among elements of candidate set. Then, the a-cut is applied to fuzzy relation T in order to transform into crisp relation as shown in (10). It is important to select a reasonable a-cut value because the hierarchical structure of candidate set depends on an applied a-cut. Finally, we draw the Hasse diagram, which completely describes a partial order among elements of candidate set, that is to say, a hierarchical structure among the elements of candidate set with respect to the optimality and efficiency. Select then the top node of the Hasse diagram as the successive heading direction of AUVs.

Because the energy consumption in vertical movement of AUVs is much greater than in the horizontal movement (1.2 times) (Ong, 1990), this technique focus strongly on the horizontal movement. In the case of obstacle occurrence, AUVs just turn left or turn right with the turning angle determined by degree from the current heading to the selected section. But in the exception case a very wide obstacle has completely filled up the sonar's coverage, AUVs must go to up one layer at a time and then apply the algorithm to find out the turning. Until obstacle clearance, AUVs are constrained to go back to the standard depth of the planned route. The algorithm of the proposed technique can summarize into five below steps and is imitated briefly in control flow as shown in Fig. 4.

Fig. 4. A control flow of collision avoidance of AUV

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