## 1 Introduction  
Robot manipulators have been used in diverse applications, including the medical, scientific, military and industrial fields. The development of fault diagnosis and fault-tolerant control (FDC) for robot manipulators is a challenging task because of the nonlinearities and coupling effects of the robot's dynamics [1]. Numerous types of failures may occur in robot manipulators, and these can be divided into three main categories: actuator faults, sensor faults and plant faults [2]. The condition monitoring of a robot manipulator can be achieved through different techniques. This research focuses on a torque and position signature analysis method, because these signals are suitable for FDC in robot manipulators. Diverse methods have been introduced for fault diagnosis and can be classified into four groups: (a) model-based techniques [2; 3], (b) signal-based approaches [4; 5], (c) artificial intelligence [6; 7] and (d) hybrid-based methods [8]. Various model-based techniques have been proposed for fault estimation (FE) in industrial components, with observation-based techniques being some of the most important algorithms. These techniques can be categorized into two main groups: linear (e.g., PI observer) and nonlinear approaches (e.g., variable structure observer (VSO), feedback linearization observer, and backstepping observer) [2; 3; 8; 9; 10]. Compared to linear observers, nonlinear observers have more edges, such as accuracy and reliability [9]. Due to the lack of robustness in the feedback linearization observers and backstepping observers, a VSO is used in this research. The VSO is reliable and robust, but is prone to high-frequency oscillations, especially in faulty conditions. These oscillations can be reduced using a high-order VSO [3; 9]. To modify the performance of high-order VSOs, various techniques have been introduced, including the quasicontinuous, high-order VSO [11], suboptimal high-order VSO [12], and the twisting high-order VSO [13]. These techniques encounter a challenge at the first-order derivative of the variable structure surface. To address this issue, a super twisting high-order VSO (HVSO) is recommended in this work. Despite its satisfactory stability, robustness and high-frequency attenuation, the fault estimation accuracy of this method must be increased to improve the rate of fault identification [9]. To increase the fault estimation accuracy and fault identification performance, a neural network algorithm is used in parallel with the HVSO to design a neural HVSO (NHVSO) for FE in the robot manipulator. The main challenge of the VSO, HVSO, and NVSO is in finding the optimal value for the variable structure surface slope in unknown (faulty) conditions. In this research, this issue is resolved using an online tuning algorithm in parallel with the NHVSO to design an adaptive NHVSO (ANHVSO). In the past decade, different machine learning solutions that utilize statistical feature parameters as attributes to learn how to solve various fault diagnosis problems, have been proposed. The most popular approaches reported in the literature include k-nearest neighbor (k-NN) classification algorithms (fault diagnosis) [14], classification and regression trees (CARTs) (fault diagnosis and prognosis) [15; 16], artificial neural networks (fault diagnosis) [17; 18], and other various types of regression algorithms (fault prognosis) [19]. However, these fault diagnosis methods have some limitations. Specifically, k-NN does not learn any specific mathematical function during its training, so the classification result is completely dependent on the quality and scale of the features used in the training set. Moreover, the value of k and the distance function used in this algorithm must be chosen, and a proper tradeoff between accuracy and the time needed for training must be found. Similarly, CART algorithms are also sensitive to the quality of the features used for training; however, they are insensitive to the scale of the data.