## 4 Cases Study
(a) Using Lemmas 2 and 4, a matrix \(G_{a}\) of the type \(G_{a}=gI,\ g>0\), which satisfies condition \(\lambda_{\min}(M^{-1}G_{a}+\left(M^{-1}G_{a}\right)^{T})=2g/\lambda_{\max}(M)\geq 2\), \(\forall p_{1}\in[0,\ 1]\), \(\forall p_{2}\in[0,\ 0.10]\) is:  
\[G_{a}=\lambda_{\max}(M)I=7.963\left[ \begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array} \right] \tag{62}\]  
(b) By using Lemmas 2 and 4, a matrix \(G_{b}\) of the type \(G_{b}=g\,diag(\left[3\,2\,1\right]),\ g>0\), which satisfies the condition \(\lambda_{\min}(M^{-1}G_{b}+\left(M^{-1}G_{b}\right)^{T})=\lambda_{\min}(M^{- 1}G_{b}+G_{b}M^{-1})\geq 2\), \(\forall p_{1}\in[0,\ 1]\), \(\forall p_{2}\in[0,\ 0.10]\) is:  
\[G_{b}=4.515\left[ \begin{array}{ccc}3&0&0\\ 0&2&0\\ 0&0&1\end{array} \right] \tag{63}\]  
(c) By using Lemmas 2 and 4, a matrix \(G_{c}\) of the type \(G_{c}=gI^{T}(\beta)\) which satisfies the condition \(\lambda_{\min}\left(JM^{-1}G_{c}+\left(JM^{-1}G_{c}\right)^{T}\right)=2g \lambda_{\min}(JM^{-1}J^{T})\geq 2\), \(\forall p_{1}\in[0,\ 1]\), \(p_{2}\in[0,\ 0.10]\) and \(\forall\beta_{1}\in[-\pi,\ \pi]\), \(\forall\beta_{2}\in[-11\pi/12,\ -\pi/12]\), \(\forall\beta_{3}\in[-11\pi/12,\ -\pi/12]\) is:  
\[G_{c}=43.994J^{T}(\beta) \tag{64}\]  
(d) By using Lemmas 2 and 4, a matrix \(G_{d}\) of the type \(G_{d}=gI^{-1}(\beta)\) which satisfies the condition \(\lambda_{\min}\left(JM^{-1}G_{d}+\left(JM^{-1}G_{d}\right)^{T}\right)\geq 2\), \(\forall p_{1}\in[0,\ 1]\), \(\forall p_{2}\in[0,\ 0.10]\) and \(\forall\beta_{1}\in[-\pi,\ \pi]\)\(\forall\beta_{2}\in[-11\pi/12,\ -\pi/12]\), \(\forall\beta_{3}\in[-11\pi/12,\ -\pi/12]\), is:  
\[G_{d}=15.516J^{-1}(\beta) \tag{65}\]  
(e) Finally, by using Lemmas 2 and 4, it can be easily verified that the constant matrix:  
\[G_{e}=5.143\left[ \begin{array}{ccc}-2.923&5.228&1.549\\ 2.184&2.326&0.683\\ 0.198&0.186&1.000\end{array} \right] \tag{66}\]  
Figure 17: Time histories of \(r_{\beta}(t)\) and \(\dot{r}_{\beta}(t)\) in the joint space. satisfies the condition \(\lambda_{\min}\Big{(}JM^{-1}G_{e}+\left(JM^{-1}G_{e}\right)^{T}\Big{)}\geq 2\), \(\forall p_{1}\in[0,\,1]\), \(\forall p_{2}\in[0,\,0.10]\) and \(\forall\beta\in I_{r_{\beta}}\), where:  
\[I_{r_{\beta}}=\big{\{}\beta:\big{|}\beta_{i}-r_{\beta_{i}}\big{|}\leq 15\pi/180, \,i=1,2,\,\,\big{|}\beta_{3}-r_{\beta_{3}}\big{|}\leq 25\pi/180\big{\}} \tag{67}\]  
In Figure 18 the neighborhood \(I_{r_{y}}\) of \(r_{y}\) in the workspace, corresponding to the neighborhoods \(I_{r_{\beta}}\) of \(r_{\beta}\), is shown. Suppose that \(M_{p3}=1.00\,\text{Kg},\,\,I_{p3}=0.10\,\text{Kgm}^{2}\). (A1) If the following a simple control law (three classic PD controllers in the joint space) is used:  
\[u(t)=7.963\left[ \begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array} \right]\Big{(}2a^{2}e(t)+2a\dot{e}(t)\Big{)},\,\,a=25,\,\,e(t) =r_{\beta}(t)-\beta(t) \tag{68}\]  
The graphical model of the majorant system \(\forall p_{1}\in[0,\,1]\) and \(\forall p_{2}\in[0,\,0.1]\), obtained with (51), is reported in Figure 19.  
From Figure 19, in accordance with Theorem 2, it is:  
if \(\|x_{0}\|_{P}\leq\rho_{1}=0.0904\), then \(\|e\|\leq\rho_{1}/a=0.0904/25=0.00362,\forallFigure 19: Function \(\dot{\rho}=f(\rho)\) and time histories of \(\rho(t)\) of the majorant system with the controller in (68). Figure 20: Time histories of the control torques with the controller (68). (A2) With the simple control law:  
\[u(t)=4.515\left[ \begin{array}{ccc}3&0&0\\ 0&2&0\\ 0&0&1\end{array} \right]\left(2a^{2}e(t)+2a\dot{e}(t)\right)\text{, }a=25\text{, }e(t)=r_{\beta}(t)-\beta(t) \tag{69}\]  
the errors \(e\) are reported in Figure 22.  
(A3) With the control law:  
\[u(t)=4.515\left[ \begin{array}{ccc}3&0&0\\ 0&2&0\\ 0&0&1\end{array} \right]\left(2a^{2}e(t)+2a\dot{e}(t)\right)-u_{g}\text{, }e(t)=r_{\beta}(t)-\beta(t) \tag{70}\]  
where \(u_{g}\) are the gravity torques, with \(a=10\), the errors \(e\) are shown in Figure 23; while with \(a=5\) the errors, as shown in Figure 24, are approximately 4 times larger, in accordance with (46).