## 2 Materials and Methods
### Fuzzy Logic Overview
#### 2.2.2 Type-2 Fuzzy Logic  
T2-FL was first defined in [PERSON] [24], but has gained significant research interest in recent years [22,29]. T2-FL is also referred to as \"General T2-FL (GT2-FL)\". A special case of GT2-FL systems is Interval T2-FL (IT2-FL). Both categories of T2-FL are parametric models with additional design degrees-of-freedom to that of a T1-FL system [30,31] and are useful in situations where determining a definitive MF is difficult [22,32]. A T2-FL set is denoted as \(\widetilde{A}\). As opposed to T1 systems, additional lower (LMF) and upper (UMF) membership functions must also be defined for T2-FL. The corresponding MFs denoted as \(\mu_{\widetilde{A}}^{L}(x)\) and \(\mu_{\widetilde{A}}^{U}(x)\) for lower and upper bounds, respectively. \(X\) now refers to the primary domain, and \(J_{x}\) is now defined as the secondary domain. Rewriting Equation (4) as a General T2 system, \(\widetilde{A}\) can be defined as (as shown in [29,33]):  
\[\widetilde{A}=\{((x,u),\mu_{\widetilde{A}}(x,u))|\forall x\in X,\forall u\in J _{x}\subseteq[0,1]\} \tag{5}\]  
What differentiates IT2-FL from GT2-FL is that the IT2-FL uses a uniform secondary MF (i.e., \(\mu_{\widetilde{A}}(x,u)=1\)), whereas the MF of GT2-FL varies with its secondary membership. As such, the resulting IT2-FL reduces the GT2-FL of Equations (5) to (6) as exemplified in Figure 4, and \(\widetilde{A}\) reduces to:  
\[\widetilde{A}=\{((x,u),1)|\forall x\in X,\forall u\in J_{x}\subseteq[0,1]\} \tag{6}\]  
Here, since the third dimensional value of the IT2-FL membership is constant, it can be more conveniently represented as a reduced two-dimensional Field of Uncertainty (FOU). For added accuracy and versatility in dealing with MF uncertainties (e.g., vehicle platform uncertainties and sensor noise), the authors choose to implement T2-FL (over T1-FL), and they choose IT2-FL (over GT2-FL) in anticipation of possible data overload burden to maintain computational feasibility, as the GT2-FL approach has been previously shown to introduce design issues and results in high computational costs [31]. As variance quantifies the uncertainty about a variable's mean in probability theory, the use of IT2-FL enables the quantification of existing uncertainties within a membership function. The FOU is defined as the area bounded by the Lower Membership Function (LMF) and the Upper Membership Function (UMF) and is depicted in the example MF in Figure 5, where the IT2-FL MF is also differentiated from that of a T1-FL. The IT2-FL FOU may be expressed as the union of all primary memberships such that  
\[FOU(\widetilde{A})=\bigcup_{\forall x\in X}(\mu_{\widetilde{A}}^{L}(x),\mu_{ \widetilde{A}}^{U}(x)) \tag{7}\]  
The mapping of the input space to that of the output space is the result of processing the input through a set of \"if-then\" linguistic rules (e.g., \"If \(x\) is \(A\), then \(u\) is \(B\)\"). The Fuzzy Inference System (FIS) begins with fuzzifying the crisp input values by using the LMFs and UMFs of the rule antecedent to determine the corresponding degree of membership in terms of linguistic metrics. This step produces two fuzzy values for each IT2-FL MF. Next, the process requires the defuzzification of the fuzzy output set to a crisp output value. The T2 output fuzzy set is reduced (via a \"type reducer\") to an Interval T1 fuzzy set resulting in a range, \(c_{L}\) (lower limit) and \(c_{R}\) (upper limit), which is considered the centroid of the T2 fuzzy set. This refers to the average of the centroids of all the type-1 fuzzy sets embedded in the type-2 fuzzy set. The centroid values are calculated iteratively due to the inability to compute exact values for \(c_{L}\) and \(c_{R}\).