## 4 Cases Study
Figure 21: Time histories of the errors with the controller (68). Figure 22: Time histories of the errors with the controller (69). If the velocity \(\hat{\beta}(t)\) is affected by a measurement noise \(n_{\hat{\beta}}\) uniformly distributed in \([-0.05,\ 0.05]\), always with the controller (70) with \(a=5\), the errors \(e\) are shown in Figure 25, in accordance with Theorem 3. Now consider the control in the workspace. (A4) With the control law:  
\[ \begin{array}{l}u(t)=43.994J^{T}(\beta)\big{(}2a^{2}e_{y}(t)+2a\dot{e}_{y}(t) \big{)}-u_{g}\\ a=5,\ e_{y}(t)=r(t)-y(t)\end{array}  \tag{71}\]  
where \(u_{g}\) are the gravity torques, the errors are shown in Figure 26.  
(A5) With the control law:  
\[ \begin{array}{l}u(t)=15.516J^{-1}(\beta)\big{(}2a^{2}e_{y}(t)+2a\dot{e}_{y}(t )\big{)}-u_{g}\\ a=5,\ e_{y}(t)=r(t)-y(t)\end{array}  \tag{72}\]  
where \(u_{g}\) are the gravity torques, the errors are shown in Figure 27.  
Figure 23: Time histories of the errors using the controller (70) with \(a=10\). Figure 24: Time histories of the errors using the controller (70) with \(a=5\). Figure 27: Time histories of the errors with the controller (72). Figure 26: Time histories of the errors with the controller (71). (A6) Finally, by using the simple control law:  
\[ \begin{split} u(t)=5.143\left[\begin{array}{ccc}-2.923&5.228&1.549\\ 2.184&2.326&0.683\\ 0.198&0.186&1.000\\ \end{array}\right]\left(2a^{2}e_{y}(t)+2a\dot{e}_{y}(t)\right)\\ e_{y}(t)=r(t)-y(t),\ a=15\end{split}  \tag{73}\]  
the errors are shown in Figure 28.  
**Remark 11**. : _Note that the control law (73) is independent of \(\beta\)._  
(B) Now consider the interaction of the considered robot with a human operator. If \(F_{e}=\left[F_{y_{1}}\ F_{y2}\ F_{y_{3}}\right]^{T}\) is the generalized force _exerted by the operator on_ the end-effector of the robot, the model is:  
\[M\tilde{\beta}=u+J^{T}F_{e}+\gamma \tag{74}\]  
Instead of opposing the action of \(F_{e}\), this action can be supported by controlling the robot such to behave as an impedance, comparable to a generalized mass-spring-damper system. The above control can be achieved using the impedance control scheme in Figure 29.  
(B1) If the considered impedance is the one of a spring-damper system (see Figure 30a), then the model is:  
\[ \begin{split}\dot{x}=-K_{a}^{-1}K_{e}x+K_{a}^{-1}F_{e}\\ \dot{\delta}_{e}=x,\dot{\delta}_{e}=-K_{a}^{-1}K_{e}x+K_{a}^{-1}F_{e} \end{split}  \tag{75}\]  
Figure 28: Time histories of the errors using the controller (73). Figure 29: Impedance control scheme. Note that the output \(\dot{\delta}_{e}\) can be used to realize the derivative action of the \(PD\) controller if \(\dot{y}\) is measurable. In the hypothesis that \(K_{e}=5I\), \(K_{a}=5/2I\), where \(I\) is the third-order identity matrix, the used controller is (73), with \(r(t)=r_{y}(t)\), assuming that the behaviors of the forces \(F_{yi}\) are as shown in Figure 31 and the behaviors of \(r_{yi}\) are as shown in Figure 32, then the behaviors of \(y_{i}\) are reported in Figure 32.  
In Figure 33, selected phases of the robot interaction with a human operator are reported assuming that the impedance is as shown in Figure 30a). Figure 31: Behaviors of the forces \(F_{yi}\). Figure 30: Virtual impedances: (**a**) spring-damper, (**b**) damper. (B2) If the considered impedance is the impedance of a damper (see Figure 30b), then the model is:  
\[ \begin{split}\dot{x}&=K_{a}^{-1}F_{e}\\ \delta_{e}&=x,\dot{\delta}_{e}=K_{a}^{-1}F_{e}\end{split}  \tag{76}\]  
In the hypothesis that \(K_{a}=5/2I\), where \(I\) is the third-order identity matrix, the used controller is (73) with \(r(t)=r_{y}(t)\), assuming that the behaviors of \(r_{yi}\) are as shown in Figure 34 and the behaviors of the forces \(F_{yi}\) are as shown in Figure 31, then the behaviors of \(y_{i}\) are reported in Figure 34.  
Figure 33: Selected phases of the interaction in the hypothesis of an impedance as in Figure 30a). (1–4 horizontal motion; 5–8 vertical motion; 9–12 angular motion).