## 2 Description of the Investigated Flexure-Based Manipulator  
The parallel flexure-based micro-motion manipulator investigated in this paper is schematically illustrated in Figure 1a and the manufactured prototype of the manipulator, which is monolithically milled from a block of Al-7075, is shown in Figure 1b. As shown in Figure 1b, the piezoelectric actuators (Pst 150/7/40, VS 12, XMT Harbin, China) adopted for the system have a 40 \(\upmu\)m range, while the output displacements of the terminal platform along the \(X\)- and \(Y\)-axes are measured by two high precision length gauges (MT-1281, Heidenhain, Bavaria, Germany). To realize high stiffness, high natural frequency and output decoupling characteristic of the micro-motion manipulator, this study used a symmetrical parallel structure composed of four identical limbs [25; 26]. Each limb is composed of a parallelogram flexure and a fixed-fixed beam, which are serially connected and acted as prismatic joints. Therefore, the micro-motion stage has two translational degrees of freedom along the \(X\)- and \(Y\)-axes. Due to the symmetrical structure of the micro-motion stage, it was necessary to study only the dynamics of the stage along the \(X\)-axis. The dynamic model of the mechanical part of the system is commonly treated as a second-order system according to the characteristics of its mechanical structure [27] and can be expressed as follows:  
\[m_{e}\ddot{x}(t)+c_{e}\dot{x}(t)+k_{e}x(t)=F_{p}(t)+f_{d}(t)+\Delta \tag{1}\]  
where \(m_{e}\), \(c_{e}\), and \(k_{e}\) are the moving mass, damping coefficient and stiffness of the stage, respectively; \(x\) is the output displacement of the terminal platform along the \(X\)-axis; \(F_{p}\) is the driving force of the piezoelectric actuator;\(f_{d}\) is the external disturbance and \(\Delta\) is the lumped uncertain of the system. As the power amplifier of the piezoelectric actuator has a very high bandwidth, it is reasonable to assume that the dynamics from the input voltage to the output displacement of the piezoelectric actuator  
Figure 1: The parallel flexure-based micro-motion manipulator. (**a**) Schematic diagram; (**b**) Manufactured prototype. can be simplified as a constant gain \(\gamma\). Thus, we can note that \(F_{p}=\gamma k_{in}u\), where \(k_{in}\) is the input axial stiffness of the micro-motion stage and \(u\) is the control input voltage. Subsequently, the dynamic model of the whole stage can be derived as a second-order system. \[\theta_{1}\ddot{x}+\theta_{2}\dot{x}+\theta_{3}x=u+\theta_{4}+d \tag{2}\]  
where \(\theta_{1}=m_{e}/(\gamma k_{in})\), \(\theta_{2}=c_{e}/(\gamma k_{in})\), \(\theta_{3}=k_{e}/(\gamma k_{in})\), \(\theta_{4}=f_{d}/(\gamma k_{in})\) and \(d=\Delta/(\gamma k_{in})\). The unknown parameter set is defined as \(\theta=[\theta_{1},\theta_{2},\theta_{3},\theta_{4}]^{\mathrm{T}}\), with an open-loop sinusoidal sweeping test subsequently being carried out to estimate the parameters in Equation (2). The sweeping sinusoidal signals ranging from 1 Hz to 1500 Hz, with the sampling frequency of 20 kHz, are used as the control input to excite the system. The system identification tool of Matlab is used to process data and estimate the model by using the response data from experiments, with the parameter estimates in \(X\)- and \(Y\)-axes direction being \(\theta_{x}=[6.35\times 10^{-7},\,5.95\times 10^{-3},\,6.23,\,0]^{ \mathrm{T}}\) and \(\theta_{y}=[5.20\times 10^{-7},\,6.10\times 10^{-3},\,6.65,\,0]^{ \mathrm{T}}\), respectively. The following practical assumption can be made. **Assumption 1:**_The extent of the parametric uncertainties and uncertain nonlinearities are known, namely:_  
\[\theta\in\Omega_{\theta}\triangleq\{\theta:\theta_{\mathrm{min}}\leq\theta \leq\theta_{\mathrm{min}}\} \tag{3}\]  
\[d\in\Omega_{d}\triangleq\{d:|d|\leq\delta_{d}\} \tag{4}\]  
_where \(\theta_{min}=[\theta_{1 min},\,\theta_{2 min},\,\theta_{3 min},\,\theta_{4 min}]^ {T}\), \(\theta_{max}=[\theta_{1 max},\,\theta_{2 max},\,\theta_{3 max},\,\theta_{4 max}]^ {T}\), and \(\delta_{d}\) are known._