# A Vibration Based Automatic Fault Detection Scheme for Drilling Process Using Type-2 Fuzzy Logic
## 4 Validation and Results
The accelerations in case of the real process can be calculated by installing accelerometer sensors on the drilling machine. Types of sensors which can be used for the process can be categorized as:  
(a) Bidirectional sensor--For the computation of \(x\) and \(y\) components of accelerations;  
(b) Axial sensor in order to calculate the acceleration along \(z\) direction;  
(c) Rotational sensor in order to calculate the acceleration along the \(\theta\) direction. \[\ddot{x}=\frac{F_{x}}{mx}+\frac{u_{x}}{m_{xx}}-\frac{c_{xx}}{m_{xx}} \dot{x} \tag{78}\]6 \(x\)-axis representation of cutting force. \[\ddot{x}=\frac{F_{x}}{mx}+\frac{u_{x}}{m_{xx}}-\frac{c_{xx}}{m_{xx}} \dot{x} \tag{78}\]7 Block diagram of drilling simulation with induced fault. \[\ddot{x}=\frac{F_{x}}{mx}+\frac{u_{x}}{m_{xx}}-\frac{c_{xx}}{m_{xx}} \dot{x} \tag{78}\]8 Block diagram of drilling simulation without fault. The entire drilling mechanism is simulated using the software platform Matlab/Simulink. The events of faults demonstrated using simulation are presented and consequently validated the fault detector scheme for successful detection of faults. For the drilling process model, two subsystem Simulink blocks are used. One block with the defect and the second block without the fault are used to compare the findings. Control signals and cutting forces are the process model's inputs. The simulation time was set for a period of 120 s. The IT2-FLS toolbox designed by [PERSON] et al. [57] is implemented in order to actuate the simulation process. The important features of the IT2-FLS toolbox are the main editor, membership function editor, rule editor as well as the surface viewer. The membership functions considered for the entire process are the Gaussian functions. Designing the type-2 fuzzy system, we first use the type-2 fuzzy logic system toolkit [57]. The computing cost of calculating the type-2 fuzzy system output is high due to its iterative nature [58]. To deal with these conditions, many TR approaches for decreasing the computational cost of the type-2 fuzzy inference mechanism have been presented. In the fuzzy logic theory, the Karnik-Mendel (KM) algorithms are iterative techniques.They are known to converge monotonically and super exponentially quickly; however, convergence requires several (typically two to six) iterations [59]. The TR methods were divided into two categories by [PERSON]: Enhancements to the KMs, which reduced the KM's computational cost, and Alternative TR approaches, which are closed-form approximations to the KM algorithm [60]. Because of its novelty and adaptability, the KM approach is the most popular [61]. A type-2 fuzzy logic system toolbox supports type reduction and defuzzification procedures: (1) Karnik-Mendel Algorithm (KM); (2) Enhanced KM Algorithm (EKM); (3) Iterative Algorithm with Stop Condition (IASC); (4) Enhanced IASC (EIASC); (5) Enhanced Opposite Direction Searching Algorithm (EODS); (6)Wu-Mendel Uncertainty Bound Method (WM); (7) Nie-Tan Method (NT); and (8) Begian-Melek-Mendel Method (BMM). It is possible to state the antecedent MFs using the MF types that currently exist in the Matlab Fuzzy Logic Toolbox in the type-2 fuzzy logic system toolbox. As a result, the LMF and UMF Matlab functions can be implemented in the same style. However, each type of MF has an additional parameter that represents the height of the related MF. A triangle MF, for example, is defined by the parameters A2, B2,C2, and H2, which specify the MF's left point, centre point, right point, and height, respectively. In type-2 fuzzy systems, especially type-2 fuzzy controller design, the parameter H2 is commonly used to produce FOU. Gaussian functions are used for the membership functions. Gaussian membership functions have the advantage of being easier to design since they are easier to express and optimise, they are always continuous, and they are faster for small rule bases.