## 1 Introduction
The proportional derivative (PD) and proportional integral derivative (PID) control, i.e., a useful control method, were employed in applications for their structural simplicity and fast responsive performance [35; 36]. However, the mentioned methods fail to sufficiently control the time-varying systems or systems exhibiting high nonlinearity [37]. As information science and technology are leaping forward, model-free adaptive control (MFAC) was proposed as a different approach to address nonlinear control issues [38; 39]. By complying with the theory of data-driven control, designers, regardless of whether the data originate from online or offline, are able to directly exploit the input/output information of the system. For practical engineering, MFAC can be designed by drawing on the knowledge from data processing without considering whether the mathematical model of the controlled system is explicit or implicit. Based on the above framework, MFAC combined with other nonlinear control methods is capable of improving the robustness of the system [40]. For instance, the scheme of the parameter's self-adjusting law and MFAC overcomes the parameter tuning process of the controller, and it is applicable to tracking the trajectory control in the PAM elbow robots [41]. Moreover, sliding mode control (SMC) acts as a useful nonlinear tool for dynamic systems exhibiting uncertain and bounded disturbances [42]. By combining the MFAC and the sliding mode exponential reaching law, a position tracking controller for a five-DOF upper-limb exoskeleton was proposed to assist victims of stroke in passive robot-assisted rehabilitation training [43]. The above attempts underpin the design of a position controller for the exoskeleton driven by PAM/PAMs with high robustness. In fact, the mentioned methods have not considered the accuracy of the tracking performance issue until the prescribed performance method was proposed [44]. The prescribed performance (PP) method is capable of ensuring the tracking error in a small predesigned boundary by appropriately defining the tracking error transformation function. Thus far, several studies in the literature have shifted from theoretically analyzing the problem of prescribed performance control to practical application. For instance, in [45], the prescribed performance control was employed in the MFAC, and the proposed method could theoretically converge the tracking error into a small boundary. In [38], a novel type of error transformation function was illustrated in the MFAC-SMC-PP framework, and the proposed method was applied in the shape memory alloy actuators to conduct practical operations. In [14], a second-order sliding mode controller integrated with the MFAC and PP method was applied for the trajectory tracking issue of the PAM upper-limb exoskeleton. However, in the mentioned controllers, the selection of parameters is dependent on the initial states, thereby indicating that the initial tracking error should be confined in the performance boundary [34]. A dilemma was identified on the application for the upper-limb exoskeleton. To be specific, once the rehabilitation training tasks have a break and should be restarted, it is required to detect the position information and select the parameters prior to the operation, thereby causing inconvenience in practice. In [34], a novel nonlinear control method was developed, and a tuning function was designed to modify the tracking error as a new variable type. Such an approach makes it possible to eliminate the limitation of the initial states without altering the stability exhibited by the system with the MFAC-SMC-PP control framework. Inspired by the mentioned observations, this study is focused on the PP method in the tracking trajectory issue of the upper-limb exoskeleton. It primarily discusses a new MFAC-SMC-PP control method to address the initial position that does not meet the performance boundary condition, which often occurs during the restart of the exoskeleton.