# LAMDA Controller Applied to the Trajectory Tracking of an Aerial Manipulator
## 3 Proposed Controllers
### LAMDA Controller
Thus, considering Equation (36), it is necessary to satisfy \(s^{T}\dot{s}\)< 0. As in the case of continuous control action, five classes are set for each input \(\dot{s}\) and \(s^{T}\). Due to the normalization of the classes that is computed \(u_{nd_{L}}\), the scaling gain \(k_{2}\) is placed at the input \(s\), and the scaling gain \(k_{d}\) at the discontinuous control output as:  
\[u_{d_{L}}=k_{d}u_{nd_{L}}\ \Rightarrow\ u_{d_{L}}=k_{d}LSMC\Big{(}s^{T},\dot{s} \Big{)};\ k_{d}>0 \tag{37}\]  
The rule table for \(u_{d_{L}}\) in which \(\dot{s}\) and \(s^{T}\) is required, considering that \(M(q(t))\) is defined positive, is shown in Table 2. The rule definition is detailed in [24]. Consider that the values presented in Table 2 for each of the classes are the ones corresponding to the weight applied to the control output \(\gamma_{k}\). Table 2 can be analyzed as follows: For example, in the class \(C_{1}\), where \(s^{T}=NB\) and \(\dot{s}=NB\), the product \(s^{T}\dot{s}\) is \(PB\). Therefore, based on Equation (36) and when \(s^{T}<0\), the negative control input is required (\(\gamma_{1}=NB\)) to quickly decrease \(s(t)\dot{s}(t)\) The same situation is presented in the classes \(C_{2},\ C_{6},\ C_{7}\). In the class \(C_{5}\), where \(s^{T}=PB\) and \(\dot{s}=NB\), the product \(s^{T}\dot{s}\) is \(NB\). Therefore, no change in the control action \(u_{d}\) is required, thus \(\gamma_{5}=ZE\). The other classes in which no change in the control action is required because the condition is met are \(C_{4},\ C_{9},\ C_{10},\ C_{16},\ C_{17},\ C_{21},\) and \(C_{22}\). As it can be seen in the controller, the \(sig()\) function used in conventional SMC controllers (see Equation (23)) is not required, since LAMDA replaces this function, making it a chattering-free controller. The overall control action is represented in Figure 5, and defined as:  
\[u_{LSMC}=k_{c}LSMC\big{(}\dot{s}\big{)}+k_{d}LSMC\Big{(}s^{T},\dot{s}\Big{)} \tag{38}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & \(\dot{s}\) & & \\ \cline{2-6}  & \(NB\) & \(NS\) & \(ZE\) & \(PS\) & \(PB\) \\ \hline \(M\) & \(C_{1}=NB\) & \(C_{2}=NS\) & \(C_{3}=ZE\) & \(C_{4}=PS\) & \(C_{5}=PB\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Rule table of LSMC for \(\dot{s}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & \(\dot{s}\) & & \\ \cline{2-6}  & \(NB\) & \(NS\) & \(ZE\) & \(PS\) & \(PB\) \\ \hline \(S^{T}\) & \(PB\) & \(C_{5}=ZE\) & \(C_{10}=ZE\) & \(C15=PS\) & \(C20=PB\) & \(C25=PB\) \\ \(PS\) & \(C_{4}=ZE\) & \(C_{9}=ZE\) & \(C14=PS\) & \(C19=PB\) & \(C24=PB\) \\ \(ZE\) & \(C_{3}=NB\) & \(C_{9}=NS\) & \(C13=ZE\) & \(C18=PS\) & \(C23=PB\) \\ \(NS\) & \(C_{2}=NB\) & \(C_{7}=NB\) & \(C12=NS\) & \(C17=ZE\) & \(C22=ZE\) \\ \(NB\) & \(C_{1}=NB\) & \(C_{6}=NB\) & \(C11=NS\) & \(C16=ZE\) & \(C21=ZE\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Rule table of LSMC for \(\dot{s}(t)\) and \(\dot{s}\). where:  
\[LSMC\big{(}\dot{s}\big{)}=\frac{\text{argmax}(\gamma_{k1})}{\sum_{k=1}^{m}\gamma_{ k1}GAD_{k,argmax(O)}}\sum_{k=1}^{m}\gamma_{k1}GAD_{k,O}\text{ \ and;}\]  
\[LSMC\big{(}s^{T},\dot{s}\big{)}=\frac{\text{argmax}(\gamma_{k2})}{\sum_{k=1}^{m }\gamma_{k2}GAD_{k,argmax(O)}}\sum_{k=1}^{m}\gamma_{k2}GAD_{k,O} \tag{39}\]  
The stability of the controller is guaranteed if selected:  
\[(k_{c}+k_{d})>\beta_{vc}-(\beta_{A}+\beta_{d}) \tag{40}\]  
With:  
\[\left\{ \begin{array}{c}\|-(M(q(t)))^{-1}C\big{(}q(t),\dot{q}(t)\big{)}\|\leq \beta_{A}\\ \\ \|\dot{\mu}_{c}(t)\|\leq\beta_{v_{c}}\\ \\ \|d\|\leq\beta_{d}\end{array} \right. \tag{41}\]  
where \(\beta_{vc}\), \(\beta_{A}\), and \(\beta_{d}\) are the limits of the bounded functions corresponding to the change of the reference, the state matrix, and the disturbance, respectively. The expression (40) guarantees \(\hat{\mathcal{L}}<0\), therefore, \(s\to 0\) with \(t\rightarrow\infty\) as proven in [26].