1. Introduction
The maxillary sinus is a conic, hollow space in the bones of the face that connects to the nose. It is the largest cavity of air in the body, and its main function is mucus production. The maxillary sinus can experience many health issues, such as nasal polyps, maxillary sinusitis, and nasal sinus cysts, which can cause infections [
1,
2]. Task complexities, narrow geometry, and limited accessibility to the sinus creates several challenges for maxillary sinus surgery. Infection of a maxillary sinus causes pain and pressure (e.g., headache and toothache). Therefore, finding a good method for maxillary sinus surgery is important [
3]. These techniques are divided into two main groups: (a) minimally invasive surgery, and (b) a non-invasive surgery technique. The creating an incision in the face, pain, and long recovery time are main challenges of the open sinus surgery. In most of cases, sinus surgery is performed as minimally invasive surgery [
4]. The natural orifice transluminal endoscopic surgery (NOTES) is one of the important techniques for minimally invasive surgery. Despite the several advantages of NOTES, surgeons still have challenges due to the narrow operating space. To improve the performance of endoscopic surgery, a number of continuum robots (e.g., concentric tube robots, tendon-driven robots, and soft robots) have been represented [
3,
4]. While concentric tube robots have several advantages, limited bending angle and small stiffness are two drawbacks of the technique [
5]. Various tendon-driven robots have been developed by various researchers for surgery [
3,
6]. A soft robot is a promising technique for surgical techniques. This technique has been used in various surgery but it is not useful for maxillary sinus surgery [
3,
7].
Figure 1 illustrates the position of the maxillary sinus. Robot-assisted surgery is used to reduce surgeon’s hand tremors, postoperative complications, and pain and to increase the precision of the actions performed during surgery. Robots have been widely used in many medical fields such as orthopedics, neurology, urology, and cardiology. Robot-assisted surgery is expanding its effects in the general medical field but has various challenges, such as non-linear behavior, the need for high-accuracy control, insertion depth perception, contact force feedback, and fault diagnosis. Various defects in robotic assistants can be categorized as actuator faults, sensor faults, and system faults. The actuator fault is affecting the continuum robot inputs. The actuator fault can be caused by abnormal operation or material aging and can drastically change the system behavior and operation. Therefore, a fault in a continuum robot motor’s joint or operator can cause the actuator fault. Continuum robots are usually equipped with sensors to recognize the surrounding environment. Unfortunately, sensors are susceptible to faults and they might lead to task failure. In this paper, actuator and sensor fault detection, diagnosis, and tolerant control are analyzed. To analyze the condition of actuator and sensor faults in robot-assisted surgery, various monitoring techniques for robot conditions such as torque, voltage, current, and vibration have been reported [
8,
9,
10]. This research exploits torque measurements because these signals are suitable for system and fault modeling, identification, estimation, and fault-tolerant control.
Various techniques have been presented recently for surgical robot fault detection, identification, and tolerant control and can be divided into two main categories: a) hardware-based techniques that rely on various physical instruments installed on the surgical robot and b) software-based techniques that utilize limited equipment. Different techniques are used for software-based fault detection, estimation, and identification, such as model-reference techniques [
11,
12], signal-based methods [
13,
14], knowledge-based algorithms [
15,
16,
17], and hybrid methods [
18,
19,
20]. Hybrid fault detection, estimation, and identification algorithms are used to design a stable and reliable technique by employing multiple methods. Different techniques have been used for hybrid methods [
18,
19,
20]. In this study, the hybrid method is designed based on three different algorithms: (a) signal-based method, (b) model-reference technique, and (c) knowledge-based algorithm. Signal-based and knowledge-based methods are used for system (surgical robotic system for the sinus) modeling based on torque and pose (position and orientation) data. A model reference technique is used to design a robust feedback linearization observer based on the variable structure technique for fault detection, estimation, and identification. A knowledge-based algorithm is used to design a Takagi–Sugeno (T–S) fuzzy technique to improve the fault estimation accuracy in the fuzzy autoregressive with exogenous input (ARX) Laguerre robust feedback linearization observer. The hybrid technique based on adaptive fuzzy observation-based estimation and feedback linearization control is used for fault-tolerant control.
Physical-based system modeling and signal-based methods for system estimation are the main methods used for modeling complex systems such as surgical robots for the sinus. Apart from the reliability of physical-based modeling of a surgical robot for the sinus, this technique has drawbacks in highly uncertain (faulty) conditions. System estimation and identification techniques such as ARX, autoregressive moving average with external input (ARMAX), orthonormal function bases (OFB), and generalized orthonormal bases (GOB) methods have been used for system estimation in various systems. Independence of the system time delay and reduction of the number of parameters are the main advantages of orthonormal techniques such as OFB and GOB with respect to classical system modeling such as ARX and ARMAX techniques. Apart from the advantages of orthonormal techniques compared to classical algorithms, these techniques have two difficulties, calculating the optimal orthonormal values and the number of restrictions in decoupled systems. To address these issues, the ARX-Laguerre technique was presented in a number of reports [
21,
22,
23]. In real world applications, the impact of noise plays a significant role in accurate system estimation. To circumvent these challenges, the fuzzy ARX-Laguerre technique is presented in this research.
Linear-based observation techniques and nonlinear-based observation algorithms are the main techniques used to design observation systems [
22,
23,
24,
25]. Linear observation systems such as proportional integral (PI) controllers have been applied in several systems for control and fault diagnosis, but their effectiveness in the presence of uncertainties is the main restriction of these algorithms [
26]. To circumvent this restriction, nonlinear-based observation techniques such as variable structure techniques [
22,
27], feedback linearization methods [
26], and fuzzy algorithms have been recommended [
28,
29,
30]. Despite the advantages of nonlinear observers, a feedback linearization observer has the challenge of limited robustness [
26], a variable structure observer has the limitation of chattering in uncertain conditions [
22], and fuzzy logic observers have the challenge of reliability and predefinition of fuzzy gain updating factors [
31] for fault detection, estimation, and identification. Consequently, a hybrid algorithm is suitable for fault detection, fault estimation, and fault identification for a surgical robot for the sinus.
Linear- and nonlinear-based fault-tolerant control techniques are the main algorithms for reducing or eliminating the effects of faults in a surgical robot for the sinus. Linear-based fault-tolerant control techniques have two main drawbacks: (a) coupling effects and (b) increase in gear ratio [
32]. However, several techniques have been introduced as nonlinear-based fault-tolerant algorithms. This technique is divided into three main categories: (a) model-based algorithms, (b) knowledge-based techniques, and (c) hybrid-based fault-tolerant control methods [
32]. Model-based and knowledge-based fault-tolerant control techniques have various positive points, but these techniques face a large challenge in the unlimited level of a faulty signal [
18,
32]. Hybrid techniques for fault-tolerant algorithms are used to address this issue [
33]. The feedback linearization algorithm can be a good candidate for a fault-tolerant control algorithm, but it must be robust. To improve robustness, the proposed observation-based feedback linearization controller is the best candidate. To increase accuracy, performance, and reliability, an adaptive, fuzzy observation-based feedback linearization fault-tolerant control algorithm is suitable for manipulators for a surgical robot for the sinus.
Figure 2 shows the block diagram of the proposed algorithm for fault detection, estimation, identification, and fault-tolerant control. The proposed adaptive ARX-Laguerre T–S fuzzy robust feedback linearization observer has the following steps: (1) modeling the surgical robot for the sinus based on the ARX method; (2) modifying the performance of the ARX technique based on an orthonormal function and designing the ARX-Laguerre method; (3) improving the accuracy of the ARX-Laguerre technique based on the fuzzy logic algorithm and designing a fuzzy ARX-Laguerre system model; (4) designing a high-performance nonlinear observer based on the fuzzy ARX-Laguerre feedback linearization observation technique; (5) improving the robustness of the fuzzy ARX-Laguerre feedback linearization observer based on the variable structure algorithm; (6) improving the performance of faulty signal estimation based on the T–S fuzzy algorithm and designing a fuzzy ARX-Laguerre T–S fuzzy robust feedback linearization observer; (7) creating a residual generation and a threshold process for fault detection and identification; (8) detecting, estimating, and identifying faults; (9) reducing the fault effect in the surgical robot for the sinus based on a feedback linearization algorithm and improving the accuracy of the fault-tolerant control using the proposed observation algorithm; and (10) tuning the feedback linearization coefficients in the various conditions and online tuning for increasing the reliability and accuracy in the proposed observation-based feedback linearization fault-tolerant control using an adaptive method.
This paper introduces three different problems and the solutions to solve these problems.
Problem 1. The main idea of the fuzzy observation-based technique for fault detection, estimation, and isolation is system modeling. Which types of system modeling are represented in this paper?
Solution 1. In the first step, the ARX technique is introduced to system modeling. To improve the robustness and reduce the effect of the noise, the Laguerre technique is used in the ARX technique as a ARX-Laguerre method. To reduce error of system estimation, a fuzzy ARX-Laguerre technique is performed for modeling of robotic maxillary sinus surgery (Section 2.2). Problem 2. Another contribution of this paper is the design of robust and reliable technique for fault detection, estimation, and isolation. Which techniques are used in this paper?
Solution 2. In this study, a feedback linearization observer is used for fault estimation. To improve the robustness of the feedback linearization observer, a variable structure algorithm is employed in the next step. To modify the fault estimation accuracy in the robust feedback linearization observer, the fuzzy technique is presented in the third step (Section 3.1). To improve the fault detection and isolation, robust residual signal and threshold value are generated, compared, and classified (Section 3.2). Problem 3. To reduce the effect of a fault in the robot surgery, which techniques are employed in this study?
Solution 3. In this study, the main idea of the fault-tolerant controller is feedback linearization control. The proposed observation technique is applied to this controller to improve the power of fault reduction. To improve the robustness in uncertain and noisy condition, a fuzzy adaptive technique is applied to the robust observation-based feedback linearization controller (Section 3.3). The rest of this research paper is organized as follows. In
Section 2, the surgical robot for maxillary sinus surgery is modeled based on the fuzzy ARX-Laguerre procedure. The proposed adaptive fuzzy ARX-Laguerre T–S fuzzy robust feedback linearization observation for surgical robot fault detection, estimation, identification, and tolerant control are presented in
Section 3.
Section 3 includes three main steps. In the first step, the fuzzy ARX-Laguerre T–S fuzzy robust feedback linearization observer based on a variable structure algorithm is utilized. In the second step, fault detection, estimation, and identification are proposed. For the fault-tolerant algorithm, an adaptive fuzzy advanced observation-based feedback linearization algorithm is recommended in the third step. In
Section 4, fault detection, estimation, identification, and tolerant control results for the surgical robot for maxillary sinus surgery are analyzed. Finally, the conclusions are provided in the last section.
2. Surgical Robot Modeling
Continuum robots have a biologically inspired form characterized by flexible backbones and high degree-of-freedom structures [
34]. These robots have the ability to move, grasp, and manipulate into tight and congested spaces. Hence, these robots have potential applications in whole-arm grasping and manipulation in unstructured environments such as those of surgery [
3]. The system information is listed in
Table 1. The bending robotic system is composed of a continuum module, system drives, and control unit. The backbone of the flexible part is obtained using a superelastic NiTiNo1 tube because of its large elastic deformation, long life span, and high performance. As a result, the manipulator is safe for surgery.
Figure 3 illustrates the continuum module of a surgical robot for the sinus.
Figure 3a illustrates the continuum joints with four holes for NiTiNo1 tube with cable (gripper) and cable (deflection). These joints are used to improve structural stability. The joints are made of stainless steel and the manipulator consists of 17 joints and each joint can be tiled up by about 30°. Based on
Figure 3b, these joints are connected by NiTiNo1 tubes and cables. Regarding
Figure 3c, these cables are used to control the scissor. To improve the operational space for the sinus robot, it is designed with the rotation and translation space by spherical joints. The rotational space is used to activate the end-effector to reach the maxillary sinus and the translation space is used to reach the front and back of the maxillary sinus. The system drives are used to activate the rotational and translational motion and the controllers are used to control of rotational, translational, deflection, and gripper (four degrees of freedom (4-DOF)) for bent to into maxillary sinus for surgical tasks [
3].
2.1. Surgical Robot Kinematics and Dynamics
As shown in
Figure 3, a surgical robot for the sinus has two segments and two deflection stages (DS).
Figure 4 shows that the proximal segment (PS) consists of areas II and III, which include seven joints, and the distal segment comprises 10 joints for areas I and IV. To analyze the continuum surgical robot for sinus deflection shown in
Figure 4, the D-H conventions are presented in
Table 2.
The translation matrix is defined by the following equation:
Here,
and
are the transformation matrix for each link, the distance from
to
measured along
, the angle from
to
measured about
, the distance from
to
measured along
, and the angle from
to
measured about
, respectively. Based on Equation (1), the forward kinematics is represented by Equation (2).
A slave robot manipulator to drive a surgical continuum robot has 4-DOF, is serial linked, and is a coupling effects system. The configurations for continuum sinus surgery with the master robot are illustrated in
Figure 5. As shown in
Figure 5, the master manipulator is used for input control, and four controllers are designed to control the four motors located in the slave manipulator. As shown in
Figure 6, these motors are used to actuate the translation, rotation, deflection, and gripper of the surgical robot for the sinus.
The dynamic formulation of the surgical robot for the sinus is represented by Equation (3) [
34,
35,
36].
where
, and
are the force coefficient matrix, inertial matrix, coefficients of the first-order generalized coordinate matrix, gravity, and faults and uncertainties, respectively. The force coefficient matrix transforms the input forces to the generalized forces and torques in the system. The inertia matrix is composed of four block matrices. The block matrices that correspond to pure linear accelerations and pure angular accelerations in the system are symmetrical. The matrix
contains coefficients of the first order derivatives of the generalized co-ordinates. Since the system is nonlinear, many elements of
contain first order derivatives of the generalized co-ordinates. The remaining terms in the dynamic equations resulting from gravitational potential energies and spring energies are collected in this matrix. Therefore, the joint position is represented by Equation (4).
Hence, the state-space equation is introduced as follows [
14]:
Here, . The , and are the state function, state input, state output, and Fourier coefficient, respectively. To design an observation-based fault diagnosis and fault-tolerant control, modeling of the surgical robot for the sinus is the first step.
2.2. Fuzzy Auto Regressive with Exogenous Input (ARX) Laguerre System Modeling
Based on
Figure 2, the proposed system modeling has three main steps: (a) ARX technique, (b) improve the robustness of the ARX technique by the orthonormal method and design of the ARX-Laguerre method, and (c) improve the accuracy of system estimation based on the fuzzy ARX-Laguerre technique. An extensive variety of sinus surgery continuum robots based on the ARX method is represented by the following equation [
37]:
Therefore, the future output can be predicted by observation parameters based on Equation (7).
where
, and
are system lag, output, input, coefficients, and zero-mean noise, respectively. Building on Equation 7, the state-space ARX modeling is represented by Equation (8).
Here,
and
are regressor variables of the ARX system modeling and the coefficient matrix, respectively. Thus, Equation (8) represents the ARX estimation technique.
Therefore, the state-space ARX system modeling is represented by the following equation.
Here,
are coefficients. To improve the accuracy and reduce the system estimation order, the ARX–Laguerre technique for modeling the surgical robot for the sinus is represented by the following equation [
18,
21].
where
and
are the Fourier coefficients, the robot manipulator order, the function of the Laguerre orthonormal, the product of the convolution, the filter system output, and the filter system input, respectively.
Figure 7 illustrates the block diagram of the ARX-Laguerre algorithm. Here,
,
,
,
,
, and
is the orthonormal basis. Therefore, the state-space ARX-Laguerre surgical robot estimation is represented by the following equation.
In this research, to modify the accuracy of the ARX-Laguerre method, a fuzzy logic technique is recommended. Based on the bounds of the uncertainties and faults, the primary objective is to present a systematic fuzzy algorithm for modeling the multi-input-multi-output (MIMO) systems in the presence of unknown parameters. Based on the fuzzy algorithm, this technique consists of the following parts: fuzzification, an IF-THEN rule, a reasoning mechanism, and defuzzification. The primary challenge of system modeling based on the fuzzy algorithm is improving the performance of the knowledge base and inference parts. The fuzzy ARX-Laguerre system estimation is defined based on the following equation:
where
is a fuzzy function for system estimation. The fuzzy IF-THEN rule is defined as follows [
28]:
where
, and
are the estimation error, the fuzzy sets, and the estimation change of error, respectively. The fuzzy membership functions for
in the interval of
are Gaussian, and the fuzzy sets are defined as negative big (NB), negative small (NS), zero (Z), positive small (PS), and positive big (PB). The fuzzy membership functions for
in the interval of
are Gaussian, and the fuzzy sets are defined as NB, NS, Z, PS, and PB. The fuzzy membership functions for
in the interval of
are Gaussian, and the fuzzy sets are defined as NB, NS, Z, PS, and PB. Thus, the state-space fuzzy ARX-Laguerre estimation for the surgical robot for the sinus is represented by:
where
are coefficients. The rule table of the fuzzy ARX-Laguerre algorithm is shown in
Table 3.
Figure 8 and
Figure 9 illustrate the estimation accuracy and errors for the normal and abnormal conditions based on the ARX method, the ARX-Laguerre technique, and the fuzzy ARX-Laguerre algorithm. According to these figures, the estimation accuracy of the fuzzy ARX-Laguerre technique is higher than those of the ARX and ARX-Laguerre techniques, and the error rate in the fuzzy ARX-Laguerre estimation technique is close to zero under normal and abnormal conditions.