## 1 Introduction  
Linear motors (LMs) have been widely applied in extremely high-precision manufacturing occasions, such as lithography machines [1], machine tools [2], and industry gantry [3, 4]. The LMs can directly produce magnetic thrust force to the load without using mechanical transmission components, such as reduction gears and lead screws [5, 6]. Meanwhile, due to the prominent advantages of high speed, large pushing force, and high precision, the studies of LMs always attract much attention from various fields, such as control engineering and industry automation. However, the control performance especially in terms of high-precision of LMs is potentially affected by various nonlinear factors, such as force ripple and friction of the LMs [7, 8, 9]. The force ripple is periodic force oscillation dependent with displacement, while the frictional force indicates interactions between the stator and mover. [PERSON] et al. present an iterative learning identification method that utilizes only partial error signals to identify the force ripple [10]. [PERSON] et al. investigate the parameters of the nonlinear friction model using particle swarm optimization and genetic algorithms [11]. [PERSON] et al. use neural networks to estimate the friction and force ripple of LMs [12]. However, implementing the above-mentioned control approach requires proper tuning for the controller gains to achieve the good performance. In addition, a heavy computational effort will increase challenges in the real-time implementation issue. To date, in the modern machining process, contour control precision for the positioning stage of multi-axis LMs has increased tremendously. Contour error is defined as the nearest distance between the actual position of the motion stage and the demanded position of the designed contour [13]. Conventionally, contour tracking performance was improved by only reducing tracking error in each axis individually. However, improving tracking accuracy only in each axis does not surely reduce contour tracking error since contour error is a geometric measurement, while tracking errors are time-related. That is, high-precision motion performance of the positioning stage of multi-axis LMs depends on both tracking and contour accuracy. To this end, the cross-coupled control methodology is widely employed to calculate the additional control action according to the tracking errors and the reference contour of each axis, and further eliminate the contouring error based on individual axial control. [PERSON] and [PERSON] employs adaptive sliding mode contouring control approach to improve machining accuracy for a biaxial computer numerical control (CNC) machine, while reducing experimentally consumed energy and control input variance [14]. In order to achieve synchronized motion performance, and while avoiding excessive internal forces for the operations of dual-drive gantry systems, [PERSON] et al. proposes a three-level synchronization control scheme based on adaptive thrust allocation, even when the load distribution on the individual axis cannot be priori known exactly [15]. [PERSON] and [PERSON] present a generalized variable-gain cross-coupled control approach to extend its applicability from just the linear contour to any contour defined by a continuously differentiable function. Their results show that the linear, circular, and parabolic trajectories can be realized well on a small two-axis CNC machine [16]. [PERSON] et al. utilize a contour algebraic equation with its partial derivatives to establish a contour error model. The error model equals zero if and only if the real contour error value vanishes, which makes perfect contour following possible, in theory [17]. Among the above-mentioned research and applications, even the cross-coupled control methodology can be improved in contouring accuracy, the model-based approach is still limited due to that the dynamical characteristics of contouring errors are unknown especially in the cases of high feed rate, large contour curvature, and sharp corner.