# Adaptive Model-Free Coupling Controller Design for Multi-Axis Motion Systems
## 1 Introduction  
In industrial applications, multi-axis motion systems have been applied many machining systems, e.g., machine tools, position stage of semi-conductor equipment, and robot manipulators. The contour error is usually adopted to evaluate the control performance; it is defined as the error component orthogonal to the desired path. For practical mechanical control systems, the contour error always exists due to mechanical structure, servo lag, as shown in Figure 1[1; 2; 3], \(e_{c}\): contour error, \(e_{uy}\): tracking error in \(y\) axis, and \(e_{ux}\): tracking error in \(x\) axis. Obviously, the tracking error is calculated every time instant and the contour one is not by instant; therefore, reducing tracking error does not effectively reduce the contour error. However, the trend of smart manufacturing is high speed and accuracy; therefore, the minimization of contour error is more important. Recently, most of industrial controllers used are PID controllers and multi-axis synchronization is less common. The key point is to design the appropriate trajectories (CNC controller interpolation, acceleration, and deceleration planning) to achieve the purpose of precision machining. Subsequently, the multi-axis control approach as proposed to treat it [4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14]. Literature [4; 5; 6] introduced the cross-coupling controller (CCC) concept to reduce the contour error. The CCC minimizes the contour error, rather than reducing the individual axis errors by the synchronization concept, and variable-gain cross-coupled controllers can instantly adjust compensator gain value based on different path models to reduce contour error, but there are restrictions on the velocity and curvature of each axis [5]. A modified variable-gain that is based on CCC scheme is proposed in [7], in which the actual contour error was replaced as a linear contour approximation to reduce the computational complexity. Recently, there are several approximated methods of contour error, and literature [8] proposed equivalent error controller design rule. The correspondingcontour error model should be constructed. Literature [9; 10] introduced three-dimensional contour error estimation methods, and a coordination transformation system was introduced to change the coordinates of multi-axis system to time-variable coordinates and calculate the contour error in time instant, as shown in Figure 2. Subsequently, an adaptive controller with linear regression property for a second order system was proposed [11]; however, the derivation of linear regression is complicated and difficult to obtain for system uncertainty. Literature [12] also utilized the local approximation of the desired path to propose a static contour error estimation method. On the other hand, dynamic estimation methods of contour error based on the Newton algorithm were introduced for achieving high precision [13; 14; 15], such approaches are typically limited by the assumptions of system dynamics should be exactly known. For a practical nonlinear plant, the system is usually not exactly known or having uncertainties or external disturbance; therefore, this motivates us to establish the adaptive model-free control method, even the controlled systems having un-modelled uncertainties and nonlinear phenomenon. Linear motors have high complexity nonlinear phenomena of friction and ripple forces. To treat this problem, literature [16; 17; 18; 19] introduced a force ripple compensation scheme based on a simple and compact model of the disturbance, and literature [20] presented three different models of friction in pre-sliding regime on linear motor platform. Herein, we propose an adaptive model-free coupling controller design while using RFNN to minimize the contour error of multi-axis linear motor system with uncertainty and nonlinear phenomenon. A time-varying contour error definition that is based on the transformation matrix is established and the design of the trajectories under the Cartesian coordinates is converted into time-varying coordinate command.