## 1 Introduction  
There are a wide variety of mechanisms such as compliant mechanism and parallel mechanism for high-precision positioning and motion technology [1; 2; 3]. Flexure-based compliant mechanisms, which transmit the motions entirely through the deformation of materials, exhibit the advantages of a typical micro-motion mechanism in terms of no backlash, no wear, and easy manufacturing [4]. Both serial and parallel structures have been adopted in flexure-based mechanisms, although most flexure-based stages are developed based on a parallel structure. This is due to the fact that parallel flexure-based stages have advantages of high rigidity and frequency, low inertia and symmetrical characteristics of each axis [5; 6; 7]. A piezoelectric actuator is usually adopted as the actuator of the flexure-based manipulator due to its merits of high-precision, fast response, high stiffness and large force [8]. However, some dynamical behaviors of micro-motion system are challenging from the perspective of control. In particular, there is inevitable model uncertainty and nonlinearity caused by cross-axis coupling effects, rigidity variation in addition to the hysteresis and other nonlinearity of the piezoelectric actuator [9; 10], which will obviously deteriorate motion accuracy, especially in high-speed applications. Therefore, the controller design plays an important role in achieving great tracking and contouring performance of flexure-based micro-motion manipulator. For general purposes, a simple proportional-integral-derivative (PID) controller is always applied in the flexure-based micro-motion stage [11]. However, due to increasing demands for higher speeds from the micro-motion manipulator, the PID controller cannot efficiently deal with the model uncertainty and nonlinear effect of the system, resulting in its limited performance. Subsequently, model-based feedforward controllers have been developed. In a previous study [12], an inversion-based feedforward controller combined with a PID feedback controller was utilized to compensate for the nonlinearity of a parallel micro-positioning stage, with good positioning and tracking performances having been achieved. Furthermore, different types of mathematical models, such as the Prandtl-Ishlinskii model [13], Presiach model [14] and Bouc-Wen model [15], have been applied for feedforward control in order to compensate the hysteresis of piezoelectric actuator. It is noted that, when the micro-motion stage is used for micro machining, the cutting forces should be considered for achieving smooth motion and high performance machining [16; 17]. However, the dynamics of the system are complicated so the accuracy of the mathematical model used in feedforward control is always limited. Furthermore, there inevitably exists model uncertainty and perturbation. For repetitive tracking tasks, iterative learning control (ILC) [18] can be adopted to compensate for the repetitive uncertainty and disturbance by using information from previous executions. In a previous study [19], ILC was used to compensate for the hysteresis of the piezoelectric actuator while too many iteration trials had to be taken in this study to get the accurate dynamical model of the system and achieve excellent tracking performance, which is too long for real-world applications. In any case, ILC is sensitive to parametric uncertainty and uncertain disturbance [18]. One typical method to deal with the model uncertainty and disturbance is adaptive robust control [20; 21; 22], which is widely used in many industrial applications. However, the velocity measurement noise greatly reduces the performance of adaptive robust control, especially in flexure-based micro-motion stages with high rigidity where velocity measurement noise is relatively large. To reduce the effect of measurement noise, a desired compensation adaptive robust control (DCARC) [23; 24] was presented.