## 2 Problem Formulation and Preliminaries
Combining the Equations (5) and (6) yields:  
\[M\ddot{\mathbf{v}}+K_{a}\dot{\mathbf{v}}=\dot{u}+\dot{\gamma}=A_{a}u+B_{a}v+C_ {a}\mathbf{v}+\dot{\gamma}=A_{a}(M\dot{\mathbf{v}}+K_{a}\mathbf{v}-\gamma)+B_ {a}v+C_{a}\mathbf{v}+\dot{\gamma} \tag{7}\]  
from which:  
\[M\ddot{\mathbf{v}}+(K_{a}-A_{a}M)\dot{\mathbf{v}}-\big{(}A_{a}K_{a}+C_{a} \big{)}\mathbf{v}=B_{a}v+\dot{\gamma}-A_{a}\gamma \tag{8}\]  
It is easy to verify that the electromechanical system (8) is of type (1) and that it can satisfy conditions (2) and (3). **Class 3. ** _Note that many robots, satellites, drones, and ships with \(m\) degrees of freedom can be represented by the equation:_  
\[M(q,p)\ddot{q}+K_{a}(q,\dot{q},p)\dot{q}+K_{e}(q,p)y=T_{u}(q)u+g(q,p)+T_{d}(q)d \tag{9}\]  
_where \(q\in R^{m}\) is the generalized coordinate vector, \(u\in R^{r}\) is the generalized control forces vector, \(g\in R^{m}\) is the generalized gravity forces vector, \(d\in R^{h}\) is the generalized disturbance forces vector, \(M(q,p)\in R^{m\times m}\) is the inertia matrix, \(K_{a}(q,\dot{q},p)\dot{q}\) is the vector of the damping, centrifugal and Coriolis forces, \(K_{e}(q,p)q\) is the vector of the possible stiffness forces, \(T_{u}(q)\in R^{m\times r}\) with \(rank(T_{u})=m\), \(T_{d}(q)\in R^{m\times h}\) are transmission matrices of the generalized forces \(u\) and \(d\), and, finally, \(p\in\wp\subset R^{\mu}\) is the vector of the uncertain parameters._  
It is easy to verify that system (9) is of type (1) and that it can satisfy conditions (2) and (3). **Class 4. ** _Consider the system:_  
\[ \begin{array}{l}\dot{x}=A(p)x+B(x,p)u+g(t,x,d,p)\\ y=C(p)x\end{array}  \tag{10}\]  
_where \(x\in R^{2m}\) is the state, \(u\in R^{m}\) is the input, \(y\in R^{m}\) is the output, \(d\in R^{h}\) is the disturbance, \(A,B,C\) are matrices of appropriate dimensions, \(g(t,x,p)\in R^{n}\) is a nonlinear vector function and \(p\in\wp\subset R^{\mu}\) is the vector of the uncertain parameters. If:_  
\[CB=0,Cg=0,rank(M_{\mathrm{O}})=2m,\mathrm{where}\ M_{\mathrm{O}}=\left[ \begin{array} []{c}C\\ CA\end{array} \right] \tag{11}\]_it is:_  
\[\dot{y}=C(p)A(p)x,x=M_{\mathcal{O}}^{-1}\left[ \begin{array}{c}y\\ \dot{y}\end{array} \right] \tag{12}\]  
\[\ddot{y} =C(p)A(p)(A(p)x+B(x,p)u+g(t,x,d,p))= \tag{13}\] \[=C(p)A(p)\left(A(p)M_{\mathcal{O}}^{-1}\left[ \begin{array}{c}y \\ \dot{y}\end{array} \right]+B\left(M_{\mathcal{O}}^{-1}\left[ \begin{array}{c}y \\ \dot{y}\end{array} \right],p\right)u+g\left(t,M_{\mathcal{O}}^{-1}\left[ \begin{array}{c}y\\ \dot{y}\end{array} \right],d,p\right)\right)=\]  
It can be readily verified that system (13) is of type (1) and it can satisfy conditions (2) and (3). **Class 5. **_Consider the transformation:_  
\[x(t)=c(q(t)),\;q\in R^{m},x\in R^{m} \tag{14}\]  
_e.g., the coordinate transformation between the joint space and the workspace of a robot or consider the nonlinear equation \(c(q)=x=\text{constant}\)._  
Let \(x(t)\) a generic trajectory \(r(t)\) with bounded second derivative, it is:  
\[\varepsilon=r-c(q),\dot{\varepsilon}=\dot{r}-J(q)\dot{q},\text{where }J(q)= \partial c/\partial q\text{ is the Jacobian matrix of }c(q) \tag{15}\]  
Moreover:  
\[\ddot{\varepsilon}=-J(q)\ddot{q}+\ddot{r}-J_{p}(q,\dot{q})\dot{q},\text{where }J_{p}(q,\dot{q})=dJ(q)/dt \tag{16}\]  
By setting \(\ddot{q}=J^{-1}(q)u,\;q=c^{-1}(r-\varepsilon),\;\dot{q}=J^{-1}\big{(}c^{-1}(r -\varepsilon)\big{)}(\dot{r}-\dot{\varepsilon})\) it can be verified that (16) is of type  
\[\ddot{\varepsilon}=-u+f(t,\varepsilon,\dot{\varepsilon},\ddot{\varepsilon}), f=\ddot{r}-J_{p}\Big{(}c^{-1}(r-\varepsilon),J^{-1}\Big{(}c^{-1}(r- \varepsilon)\Big{)}(\dot{r}-\dot{\varepsilon})\Big{)}J^{-1}\Big{(}c^{-1}(r- \varepsilon)\Big{)}(\dot{r}-\dot{\varepsilon}) \tag{17}\]  
Hence the model of the inversion error of the transformation \(x(t)=c(q(t))\) is of type (1), satisfies condition (2), and it can satisfy condition (3).