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.

本文基于稠密集（TSs）和渐进集（GSs）的概念，提出了一种新的概念：稠密渐进集（TGS）。TS是不确定集合，由一对明快集合（CS）表示。这些CS表示TS的上限和下限。因此，可以将TS视为CS的间隔。GS是CS，通过相关程度进行参数化，旨在提高CS的特异性。此外，TGS是GS的间隔，即一对下GS和上GS。在保证单调性（一致性）约束的情况下，GS变为1型模糊集（T1FS），而TGS可以视为厚模糊集（TFS）。此外，由T1FS上下边界组成的TFS可以解释为2型模糊集（T2FS）。根据TGS的表示，这种新方法为解释，操纵和计算一些不确定量提供了一个原始概念，这些不确定量不能用GS，T1FS和/或T2FS表示。TGS概念的潜在应用已在解决方程式模糊系统和不确定性模糊回归的框架中使用应用实例进行了验证，并通过实际应用进行了验证，其中水下机器人的轨迹不确定并且由于干扰而无法精确知道由环境引起的。所提出的方法使得计算水下机器人探索的不确定区域成为可能。TGS概念的潜在应用已在解决方程式模糊系统和不确定性模糊回归的框架中使用应用实例进行了验证，并通过实际应用进行了验证，其中水下机器人的轨迹不确定并且由于干扰而无法精确知道由环境引起的。所提出的方法使得计算水下机器人探索的不确定区域成为可能。TGS概念的潜在应用已在解决方程式模糊系统和不确定性模糊回归的框架中使用应用实例进行了验证，并通过实际应用进行了验证，其中水下机器人的轨迹不确定并且由于干扰而无法精确知道由环境引起的。所提出的方法使得计算水下机器人探索的不确定区域成为可能。