## 3 Control System Design for the Flexure-Based Manipulator
### Reference Optimization in LODCARC  
The proposed DCARC term in Section 3.1 is designed to track \(x_{a}=x_{opt}+x_{d}\) as accurately as possible, which also means if \(x_{opt}=0\), DCARC term can track the desired reference \(x_{d}\) as accurately as possible. Furthermore, Theorem 1 reveals the stability of the result, even resulting in a zero tracking error. However, in real-world applications, tracking error is inevitable (i.e., \(e_{a}=x-x_{a}\) cannot be zero) and in practice, there exists residual tracking error of the practical motion control system \(e_{a}\). As \(e_{a}=x-(x_{opt}+x_{d})=e-x_{opt}\), we have  
\[e=e_{a}+x_{opt} \tag{28}\]  
Thus, if the optimal input signal \(x_{opt}\) could capture the characteristics of \(-e_{a}\) (i.e., \(x_{opt}\to e_{a}\)), the actual tracking error \(e\) could achieve \(e\to 0\). Therefore, iterative learning control (ILC), which can improve tracking performance by using information from previous executions to compensate for the repetitive uncertainty and perturbation, is used to capture the characteristics of the residual tracking error. As shown in Figure 2, the ILC term is used to generate the optimal input signal for adjusting the desired reference of DCARC. Denote the dynamics of the closed-loop of DCARC with the flexure-based plant as \(P\)(s), then the whole control system can be derived as  
\[X_{i}(s)=P(s)[X_{opt,i}(s)+X_{d}(s)] \tag{29}\]  
where subscript \(i\) denotes the iteration trial number, while the optimal input signal \(X_{opt,i}\) is generated by a iterative learning law as  
\[X_{opt,i}(s)=Q(s)(X_{opt,i-1}(s)+L(s)E_{i-1}(s)) \tag{30}\]  
where \(E_{i-1}\) is the tracking error in the \((i-1)\)th iterative process, \(Q\) is a low-pass \(Q\)-filter which is utilized to enhance the system robustness and suppress the noise in the iterative process, while \(L\) is the learning function. It is noted that a low-pass filter may cause a phase shift, although the phase shift can be eliminated by filtering the signal back and forth [29]. Many types of learning functions have been developed for ILC [18], while PD-type learning function is a typical, simple and tunable ILC learning function. PD-type learning function was chosen for this paper and it is written as  
\[L(s)=K_{p}+K_{d}s \tag{31}\]  
Following this, asymptotic stability and monotonic convergence in the iteration domain of the iterative law can be guaranteed under certain conservative conditions, which are presented in [18] as the following Lemma.  
**Lemma 1:**_The iterative learning law defined in Equations (30) and (31) acting on the system P(s) is monotonically convergent i.e.,\(\left\|E_{\infty}(s)-E_{i+1}(s)\right\|_{\infty}<\alpha\left\|E_{\infty}(s)-E_{i}(s) \right\|_{\infty}\), where \(\alpha\) is the convergence rate, if_  
\[\left\|Q(s)(1-L(s)P(s))\right\|_{\infty}<\alpha<1 \tag{32}\]  
_Based on Lemma 1, the optimal input signal \(x_{\text{opt}}\) generated by ILC control can capture the characteristics of residual tracking errors caused by unmodeled repetitive uncertainty. If the convergence of ILC is guaranteed, the optimal input signal \(x_{\text{opt}}\) will be determined and then according to Theorem 1, the stability of the whole control system can be guaranteed. Furthermore, as mentioned, zero tracking error is achieved if there are only parametric uncertainties so P(s) can be considered as P(s) = 1 theoretically. After this, the convergence conditions can be obtained by substituting Equation (31) into Equation (32) to obtain_  
\[\left\|Q(s)(1-K_{p}+K_{d}s)\right\|_{\infty}<\alpha<1 \tag{33}\]  
_Obviously, if \(K_{p}\) + \(K_{d}s\) = 1 (i.e., \(K_{p}\) = 1 and \(K_{d}\) = 0), then we could obtain \(\alpha\) = 0 in theory, which means the ILC term will reach a converged error value after one iteration._