Thick gradual sets and their computations: Application for determining the uncertain zone explored by an underwater robot. (June 2021) - Boukezzoula, Reda; Jaulin, Luc; Desrochers, Benoit; Foulloy, Laurent
Abstract: This paper proposes a new concept of thick gradual sets (TGSs), which is based on the notions of thick sets (TSs) and gradual sets (GSs). A TS is an uncertain set, which is represented by a pair of crisp sets (CSs). These CSs represent the upper and lower bounds of the TS. Therefore, a TS can be considered as an interval of CSs. A GS is a CS, which is parameterized by a degree of pertinence and aims to increase the specificity of CSs. Furthermore, a TGS is an interval of GSs, i.e., a pair of lower and upper GSs. In situations when the constraint of monotonicity (consistency) is guaranteed, a GS becomes a type-1 fuzzy set (T1FS) and a TGS can be regarded as a thick fuzzy set (TFS). Moreover, a TFS, which is composed of lower and upper T1FS bounds, can be interpreted as a type-2 fuzzy set (T2FS). According to the TGS representation, this new approach offers an original concept for interpreting, manipulating, and computing some uncertain quantities that cannot be represented by GSs, T1FSs, and/or T2FSs. The potential applications of the TGS concept has been validated using application examples in the frameworks of solving fuzzy systems of equations and uncertain fuzzy regression and through a real-world application where the trajectory of an underwater robot is uncertain and cannot be precisely known because of disturbances induced by the environment. The proposed approach makes it possible to compute the uncertain zone explored by the underwater robot.