## 3 Control System Design for the Flexure-Based Manipulator
### DCARC Design in LODCARC  
As mentioned, DCARC has a parametric adaptation ability of adaptive control and having robust performance to the uncertainty and disturbance of robust control. The design of the DCARC term of the proposed LODCARC for the flexure-based system is as follows: Define \(\hat{\theta}\) as the estimate of \(\theta\), then  
Figure 2: Structure of the control system for the flexure-based micro-motion manipulator. the estimation error of \(\theta\) can be defined as \(\widetilde{\theta}=\hat{\theta}-\theta\). In the DCARC term, the estimate \(\hat{\theta}\) of parameter is updated by using the following discontinuous projection type adaptation law:  
\[\dot{\hat{\theta}}=\text{Proj}_{\hat{\theta}}(\Gamma\tau) \tag{5}\]  
where \(\Gamma\) is a diagonal positive definite matrix, which indicates the adaptation rate, and \(\tau\) is an adaptation function. The projection mapping \(\text{Proj}_{\hat{\theta}}(\bullet)=\left[\text{Proj}_{\hat{\theta}_{1}}( \bullet_{1}),\cdot\cdot\cdot,\text{Proj}_{\hat{\theta}_{4}}(\bullet_{4}) \right]^{\text{T}}\) is defined by  
\[\text{Proj}_{\hat{\theta}_{i}}(\bullet_{i})=\left\{ \begin{array}{ll}0,\text { if }\hat{\theta}_{i}=\theta_{\text{max}}\text{ and }\bullet_{i}>0\\ 0,\text{ if }\hat{\theta}_{i}=\theta_{\text{min}}\text{ and }\bullet_{i}<0\\ \bullet_{i},\text{ otherwise}\end{array} \right. \tag{6}\]  
It can be shown [28] that for any adaptation function \(\tau\), the projection mapping defined in Equation (6) guarantees  
\[ \begin{array}{ll}\text{P1}:&\hat{\theta}\in\Omega_{\theta}\triangleq\left\{ \hat{\theta}:\theta_{\text{min}}\leq\hat{\theta}\leq\theta_{\text{max}}\right\} \\ \text{P2}:&\widetilde{\theta}^{\text{T}}(\Gamma^{-1}\text{Proj}_{\hat{\theta}} (\Gamma\tau)-\tau)\leq 0,\forall\tau\end{array} \right.. \tag{7}\]  
After this, define a switching-function-quantity as  
\[p=\dot{e}_{a}+k_{1}e_{a}=\dot{x}-x_{eq},\ x_{eq}\triangleq\dot{x}_{a}-k_{1}e _{a} \tag{8}\]  
where \(e_{a}=x-x_{a}=x-(x_{d}+x_{opt})\) is the adjusted tracking error; \(x\), \(x_{d}\) and \(x_{a}\) are the actual output, desired reference and adjusted optimized reference, respectively; And \(k_{1}\) is a positive feedback gain. Differentiating Equation (8) and noting Equation (2), one can obtain  
\[ \begin{array}{ll}\theta_{1}\dot{p}&=u-\theta_{1}\dot{x}_{eq}-\theta_{2}\dot {x}-\theta_{3}x+\theta_{4}+d\\ &=u+\varphi^{\text{T}}\theta+d\end{array}  \tag{9}\]  
where \(\varphi=\left[-\dot{x}_{eq},-\dot{x}_{a},-x_{a},1\right]^{\text{T}}\). Define \(\varphi_{a}=\left[-\ddot{x}_{a},-\dot{x}_{a},-x_{a},1\right]^{\text{T}}\) and the DCARC control law is proposed as  
\[u=u_{f}+u_{s} \tag{10}\]  
where \(u_{f}=-\varphi_{a}^{\text{T}}\hat{\theta}\) is the adjustable model compensation term for perfect trajectory tracking. It should be noted here that the adjustable model compensation is decided by adjusted optimized reference and parameter estimations only, rather than the actual output, which has great velocity measurement noise. In this way, DCARC can minimize the effect of measurement noise compared with conventional adaptive robust control. Furthermore, \(u_{s}\) is the feedback control term, which will be synthesized later. Substituting Equation (10) into Equation (9) and then simplifying the resulting expression leads to  
\[\theta_{1}\dot{p}=u_{s}-\varphi_{a}^{\text{T}}\widetilde{\theta}+d+(k_{1} \theta_{1}-\theta_{2})\dot{e}_{a}-\theta_{3}e_{a}. \tag{11}\]  
In addition, the feedback control term \(u_{s}\) in Equation (10) consists of two terms given by  
\[u_{s}=u_{s1}+u_{s2} \tag{12}\]  
where \(u_{s1}=\text{-}k_{s1}p\) is used to stabilize the nominal system and \(u_{s2}\) is a robust feedback used to attenuate the effect of model uncertainties. For traditional adaptive robust control, \(u_{s1}\) is just a simple proportional feedback with \(k_{s}\) being the feedback gain to stabilize the nominal system [24], while in this case for DCARC, \(k_{s1}\) in Equation (12) is required to be large enough such that the matrix \(A\) defined below is positive definite. \[A=\left[