## 4 Structure of the First Diagram for Robot Control, the Iterative Learning Method
### Model-Free Disturbance Compensation for Internal Loop Control by Feedback Linearization  
Going back to the pay provider \(\underline{\underline{\theta}}_{k}\) in Equation (16) and designating as the present with \(t=kT+\tau;0\leq\tau<T\) and \(\underline{\underline{\theta}}_{k}=\underline{\underline{\theta}}_{k}(\tau); \underline{\underline{X}}_{k}=\underline{\underline{X}}_{k}(\tau)\) then (17) is rewritten as:  
\[\underline{\underline{X}}_{k}(\tau)=A\underline{\underline{X}}_{k}(\tau)+B \Big{[}\underline{\underline{\theta}}_{k}(\tau)-\underline{\underline{\theta }}_{k}(\tau)+\underline{\underline{\mu}}_{k}(\tau)\Big{]} \tag{19}\]  
The automatic model-free feedback linearization block (Figure 2) aims to estimate \(\underline{\underline{\eta}}\) based on the measured values of \(\underline{\underline{X}}_{k}(\tau)\) at two times \(\tau=nt_{s}\) and \(t-t_{s}=t_{s}(n-1)\) for each corresponding \(\underline{\underline{X}}_{k}(nt_{s});\underline{\underline{X}}_{k}(n-1)t_{s}\) value. With \(0<t_{s}<<1\) being an arbitrarily chosen constant small value, we then expand the Taylor function of the function \(\underline{\underline{X}}_{k}(\tau)\) around \((n-1)t_{s}\), as follows:  
\[\underline{\underline{X}}_{k}((n-1)t_{s})=\frac{t_{s}^{2}}{2}\underline{ \underline{\chi}}_{k}(\varsigma)-t_{s}\underline{\underline{\chi}}_{k}(nt_{s} )+\underline{\underline{\chi}}_{k}(nt_{s}) \tag{20}\]  
where \((n-1)t_{s}\leq\varsigma\leq nt_{s}\),_or  
\[\underline{\underline{\chi}}_{k}(nt_{s})\approx\frac{\underline{\underline{ \chi}}_{k}(nt_{s})-\underline{\underline{\chi}}_{k}((n-1)t_{s})}{t_{s}}. \tag{21}\]  
Assuming the last term of expression (20) is very small and can be ignored, the expression \(\underline{\underline{\eta}}_{k}(nt_{s})\) will be used to approximate Equation (19) by replacing the signs \"\(\approx\)\" and \(\underline{\underline{\eta}}_{k}(nt_{s})\) in (10) with \"\(=\)\" and \(\underline{\underline{\eta}}_{k}(nt_{s})\)  
\[\frac{\underline{\underline{\chi}}_{k}(nt_{s})-\underline{\underline{\chi}}_{ k}((n-1)t_{s})}{t_{s}}\approx A\underline{\underline{\chi}}_{k}(nt_{s})+B\Big{[} \underline{\underline{\sigma}}_{k}(nt_{s})-\underline{\underline{\underline{ \eta}}}_{k}((n-1)t_{s})+\underline{\underline{\mu}}_{k}(nt_{s})\Big{]} \tag{22}\]  
Then, we calculate:  
\[B\underline{\underline{\eta}}_{k}(nt_{s})=\frac{\underline{\underline{\chi}}_{ k}(nt_{s})-\underline{\underline{\chi}}_{k}((n-1)t_{s})}{t_{s}}-A\underline{ \underline{\chi}}_{k}(nt_{s})-B\Big{[}\underline{\underline{\theta}}_{k}(nt_ {s})-\underline{\underline{\underline{\eta}}}_{k}((n-1)t_{s})\Big{]}\]Which yields:  
\[\underline{\underline{\eta}}_{k}(nt_{s})=B^{T}\bigg{[}\frac{\underline{X}_{k}(nt_{s })-\underline{X}_{k}((n-1)t_{s})}{t_{s}}-A\underline{X}_{k}(nt_{s})\bigg{]}- \Big{[}\underline{g}_{k}(nt_{s})-\underline{\underline{\eta}}_{k}((n-1)t_{s}) \Big{]} \tag{23}\]  
Theorem 1: _The value \(\underline{\underline{\eta}}_{k}(nt_{s})\) calculated in Equation (23) causes the approximation error of expression (22) to decrease._  
Proof: Error in expression (22) \(\underline{\varepsilon}\) will be calculated:  
\[ \begin{array}{ll}\underline{\sigma}&=A\underline{X}_{k}(nt_{s})+B\Big{[} \underline{\underline{\theta}}_{k}(nt_{s})-\underline{\underline{\mu}}_{k}(( n-1)T_{s})+\underline{\underline{\eta}}_{k}(nt_{s})\Big{]}-\frac{\underline{X}_{k}(nt_{s })-\underline{X}_{k}((n-1)t_{s})}{t_{s}}\\ &=B\underline{\underline{\mu}}_{k}(nt_{s})+\underline{\underline{\lambda}} \end{array}