## 4 Fuzzy Control System  
In this study, controls for rotational motion (i.e., roll, pitch, and yaw) were based on the fuzzy control system, as illustrated in Figure 7. The design process for the input, fuzzy inference, and fuzzy rule base is detailed as follows:  
Figure 6: Definitions of object-tracking values of (**a**) benchmark, (**b**) orientation, and (**c**) distance.  
Figure 7: Fuzzy control system structure.  
1. Definitions of input and output variables  
Because of the cruciform rudder's design, the rudder control can be divided into horizontal and vertical rudder planes. The controls on the horizontal and vertical rudder planes manage pitch and the yaw motions, whereas the revolution speed of the propeller controls thruster force.  
2. Fuzzification strategy determination  
The present study defined an input variable of a target in the Y-direction of the visual plane and an output variable for the rudder angle. Figure 8a illustrates the target in the visual coordinate as an input variable and defines the five following fuzzy values: Strong Left, Left, Middle, Right, and Strong Right. In addition, Figure 8b defines the rudder angle as an output variable ranging between \(-30^{\circ}\) and \(+30^{\circ}\), and the five fuzzy values are Strong Left, Left, Keep, Right, and Strong Right.  
Figure 9a defines the target in the X-direction of the visual coordinate plane as an input variable and defines five fuzzy values, namely Strong Up, Up, Middle, Down, and Strong Down. In addition, Figure 9b displays the vertical rudder angle as an output variable with a range between \(-30^{\circ}\) and \(+30^{\circ}\), and the five fuzzy values are Strong Up, Up, Keep, Down, and Strong Down.  
When the size of an input variable for the target object on the visual plane is defined, propeller revolution speed is regarded as an output variable, as presented in Figure 10. Figure 10a presents the input variable of the target object in the visual coordinate plane with nine defined fuzzy values: SSmall, VSmall, Small, LSSmall, Middle, LBig, Big, VBig, and SBig. Figure 10b presents the propeller speed as an output variable ranging between 0 and 2000 rpm and nine fuzzy values: SFast, VFast, Fast, LFast, Keep, LSlow, Slow, V Slow, and STOP.  
3. Design of the fuzzy control rule base  
The fuzzy control rules in this study were developed on the basis of the AUV experiments. Tables 2-4 present the rules for the vertical rudder plane, horizontal rudder plane, and thruster revolution speed, respectively.  
4. Fuzzy inference  
The decision calculus was based on the fuzzy rule base. The system simulates human reasoning through this process called fuzzy inference. This study was based on t-conorm and t-norm methods [27]. T-conorm is a union operation between the IF and OR operation partial premises, whereas t-norm is an intersection operation between the IF and operation partial premises.  
5. Defuzzification selection  
Defuzzification converts the results of fuzzification obtained from reasoning into a crisp set. It is based on the weighted average formula (Equation (2)), where \(\mu_{i}(y)\) is the membership function of the output set, \(\alpha_{i}\) the weight of the \(i\)th rule, and \(N\) the total number of rules. Equation (3) represents the center of area, Equation (4) the center of sums, and Equation (5) the mean of maximum defuzzification--all of which are derived from the weighted average formula.  
\[y(x)=\sum_{i=1}^{N}\alpha_{i}\mu_{i}(y)/\sum_{i=1}^{N}\alpha_{i} \tag{2}\]  
\[y=\frac{\sum_{i=1}^{N}y_{i}\mu_{i}(y_{i})}{\sum_{i=1}^{N}\mu_{i}(y_{i})} \tag{3}\]  
\[y=\frac{\sum_{i=1}^{N}y_{i}\sum_{j=1}^{N}y_{j}\mu_{j}(y_{i})}{\sum_{i=1}^{