## 1 Introduction
To solve the control problem of multi-axis LMs, the proportional-integral-derivative (PID) control has been popularly adopted in industrial applications for several years due to its simple structure and easy implementation. However, it is difficult to obtain satisfactory control performance in nonlinear and time-varying systems using the conventional PID controller since it is not robust enough to achieve satisfactory control performance [18]. As a relief to this problem, many advanced PID control strategies have been developed, such as two-degree-of-freedom PID (2 PID) [19], internal model controller PID [20], automatic tuning PID [21], and fractional-order PID [22]. Among these improved PID control methods, a 2 PID controller acquires a better response as compared to that of a conventional one-degree-of-freedom PID (PID) controller due to its additional control loop [23]. However, designing a 2 PID controller is more complicated than that of a PID controller. With the aid of fuzzy logic control, [PERSON] et al. establish a hybrid method based on the 2 PID controller with a first-order time-delay system, in which the parameters can be automatically tuned to take advantages of reference tracking and disturbance rejection operation [24]. [PERSON] et al. adopt a single layer neural network with a 2 PID controller to change the flow rate of a cooling agent in a fermentation bioreactor. The results show that the 2 PID controller performs better performance than a PID controller due to the additional controller parameters [25]. [PERSON] et al. propose a resonant-2 PID controller that combines the quasi-resonant controllers and a 2 PID controller to suppress current harmonics for permanent magnet synchronous linear motors [26]. The resonant controller is added in parallel to the PI controller to suppress the current harmonic components, and a 2 PID controller is additionally adopted to effectively reduce the raised overshoot in the current loop response. However, while applying these methods on the multi-axis LMs, the number of the 2 PID controller parameters increases rapidly and leads to more design complexity and is time-consuming during the process in designing and optimizing these parameters. Population-based evolutionary algorithms (EAs) are commonly used to find near-optimal solutions. Among several EAs, the optimization method based on the artificial bee colony (ABC) algorithm is inspired by the collective foraging behavior of honey bee colonies [27; 28]. Since the information about the nectar amount available in food sources will be shared among honey bees, the location of food source with more nectar can be regarded as closer to the optimal food source. To mimic such behavior of honey bees, in the ABC algorithms, each food source location can represent a possible solution and the optimization problem is considered to be the finding of optimal food source that implies the optimal solution [28]. According to the comparative study in Reference [29], even with fewer control parameters the ABC algorithm can further outperform the other EAs in terms of several advantages of simplicity, robustness, high flexibility, and ease of real-time implementation. To this end, the ABC algorithm is widely applied in recent engineering optimization problems, such as DC motor drive [30], controller optimization [31], and learning algorithm for humanoid robots [32]. On a different note, this study thus takes the challenge to propose a control strategy of auto-tuning cross-coupled 2 PID (ACC2 PID) for the mover positioning on an LMs-driven X-Y-Y stage with extremely high-precision control performances in both position tracking in each axis and contouring in biaxial motions. Firstly, the dynamical characteristics of X-Y-Y stage using three LMs are derived, and the corresponding model of contouring error is established.