Learning Based Cross-coupled Control for Multi-axis High Precision Positioning Systems

In this paper, a controller featuring cross-coupled control and iterative learning control schemes is designed and implemented on a modular two-axis positioning system in order to improve both contour and tracking accuracy. Instead of using the standard contour estimation technique proposed with the variable gain cross-coupled control, a computationally efficient contour estimation technique is incorporated with the presented control design. Moreover, implemented contour estimation technique makes the presented control scheme more suitable for arbitrary nonlinear contours. Effectiveness of the control design is verified with simulations and experiments on a two-axis positioning system. Also, simulations demonstrating the performance of the control method on a three-axis positioning system are provided. The resulting controller is shown to achieve nanometer level contouring and tracking performance. Simulation results also show its applicability to three-axis nano-positioning systems. INTRODUCTION Increasing demand for micro/nano-technology related equipment resulted in a growing interest for precision positioning. Multi-axis precision positioning is required in micro/nano-scale manufacturing and assembly, optical component alignment systems, scanning microscopy applications, nano-particle placement applications, cell/tissue engineering and etc. [1-3]. Most of the time, these applications require both high contour and tracking accuracy. In tracking control, the objective is moving along a desired trajectory. Although almost all systems employ feedback as a part of tracking control, substantial improvement of tracking accuracy is achieved by the addition of feed forward control methods. In literature, several feed forward control schemes have been shown to improve tracking accuracy such as zero phase error tracking control (ZPETC) [4-6], feed forward friction compensation [7, 8] and iterative learning control (ILC) [9, 10]. According to Tomizuka [4], tracking performance of a ZPETC system is sensitive to variations in plant parameters and modeling errors since ZPETC design is based on pole/zero cancellation and phase cancellation. Moreover, friction compensation techniques generally incorporate a system identification process that should be repeated if system parameters change. On the other hand, Tan et al. [9] claims that specifying a plant model for ILC via zero phase filtering is not necessary considering the principle of self-support that is argued in [11] because the stored control signals reflect the plant characteristics. In other words, ILC can improve tracking performance of a system even the plant structure and nonlinearities are unknown [12]. Yet, the system should execute the same task repetitively to be able to implement an ILC scheme. Generally, improving tracking accuracy of each individual axis also increases contouring accuracy of the multi-axis system. However, in some cases, decreasing the tracking error may not decrease the contour error; it may even deteriorate the contouring performance [13]. Hence, control structure should be designed considering not only tracking error but also contour error in order to achieve high accuracy in both. Koren [14] proposed the cross-coupled control (CCC) that focuses on eliminating contour error rather than individual axes errors. This method is proven to reduce contour error significantly. Since the introduction of CCC, it has been modified and combined with different control techniques. Some examples are observer-based CCC [15], cross-coupled model reference adaptive control [16], cross-coupled iterative learning (CCILC) [10], CCC with disturbance observer and ZPETC [6], CCC with friction compensation [8] and CCC with ILC [10, 17]. Since CCC based control schemes require contour error as the control parameter, there is a need for construction of a contour error model in real time. Contour error is defined as distance between actual position and the nearest position on the contour [18]. Although, contour error can be calculated for linear contours, this calculation is very complicated for nonlinear contours, especially during the operation. Hence, some approximations have been used to calculate a nonlinear contour error. Koren [13] suggested circular contour assumption. Then, Yeh and Hsu [18] proposed a method to approximate contour error as the vector from the actual position to the nearest point on the line that passes through the reference position tangentially. As the authors mentioned, although circular contour assumption works well for biaxial motion systems, it is difficult to apply on multi-axis systems. The work presented aims to provide an improved method for precision motion control featuring CCC and ILC. Although CCC and ILC have been used together in [10] and [17] for contours combining lines and circles, the new method also benefits from the contouring error estimation vector approach. In this way, the new method is computationally more efficient, more suitable for coupling gain calculations of arbitrary nonlinear contour and easier to implement on multi-axis systems. Moreover, for the best of our knowledge, this is the first time CCC and ILC is used together to achieve nanometer level precision and implemented on a three-axis system. SYSTEM SETUP The two-axis positioning system is constructed by assembling two modular single-axis stages perpendicularly as in Fig. 1. A modular single-axis stage is designed with a stationary base and a moving slider that are connected to each other via cross-roller linear bearings. The stage is actuated by a brushless permanent magnet linear (PMLM) motor with 120mm travel range whereas the position feedback is taken from an incremental linear encoder. The linear encoder has an optical scale with four micrometer grating in pitch leading one micrometer resolution. Yet, the encoder resolution is increased to 25 nanometers using an interpolation technique. Details of interpolation procedure can be found in [19]. Idealized dynamic model of a single-axes linear stage is given in Fig. 2 where R is linear motor resistance, L is linear FIGURE 1. TWO-AXIS MODULAR POSITIONING SYSTEM motor inductance, KBEMF is back electromotive constant, Kforce is force constant, m is sliding mass, b is viscous friction, e is linear motor input voltage, Kamp is amplifier gain and i is linear motor current. In the dynamic model, ripple forces of the PMLM are neglected and linear bearings are modeled as viscous friction component. From the dynamic model, mathematical model of the linear stage is found as in Eq. (1).   2 ( ) ( ) ( ) ( ) ( ) amp force BEMF force K K X s G s E s s Lms Rm bL s Rb K K       (1) After the system is modeled, a suitable PID feedback controller is obtained using traditional methods. FIGURE 2. DYNAMIC MODEL OF A SINGLE-AXIS STAGE CONTROL DESIGN In this paper, an improved method based on CCC and ILC which benefits from the contouring error vector approach has been presented. Next two subsections will briefly describe ILC scheme used in this work and CCC. Moreover, contour estimation approaches will be explained together with CCC. The last subsection will mention the insight of the improved method. Iterative Learning Control (ILC) via Zero Phase Filtering ILC is a technique for improving the transient response of a system that operates repetitively. ILC can often be used to achieve perfect tracking, even when the model is uncertain or unknown and there is no information about the system structure and nonlinearity [12]. ILC based on zero phase filtering is a practical and efficient implementation of ILC [9]. Block diagram of ILC via zero phase filtering for an individual axis is given in Fig. 3. In the diagram, superscript i is iteration number whereas uff i and ufb i are feed forward and feedback control signals at i th iteration. The feed forward control signal for i th iteration is calculated using the feed forward and feedback control signals of the previous iteration that are shown as uff i-1 and ufb i-1 respectively. The learning update law can be given as in Eq. (2) [9]. 1 1 ( ) ( ) ( ) 2 1 M i i i