## 3 Dynamic Modelling of the Hybrid Robots Class III
By using the following Lemma, \((n+2m)\) differential equations of second order, Equation (10), can be transformed into only \(n\) differential equations of second order which are expressed in terms of only \(n\) independent generalized coordinates, not including \(2m\) Lagrangian Multipliers \(\lambda=\left[ \begin{array}{cccc}\lambda_{1}&\lambda_{2}&\ldots&\lambda_{2m }\end{array} \right]^{T}\). **Lemma. **_Consider a generalized hybrid robot mechanism Class III which has \(n\) independent joint variables \(\mathbf{u}=\left[ \begin{array}{cccc}u_{1}&u_{2}&u_{3}&\ldots&u_{n}\end{array}  \right]^{T}\) and \(2m\) dependent joint variables \(\mathbf{z}=\left[ \begin{array}{cccc}z_{1}&z_{2}&\ldots&z_{2m}\end{array}  \right]^{T}\). The relationship between \(\mathbf{u}\) and \(\mathbf{z}\) is expressed as \(\dot{\mathbf{s}}=\left[ \begin{array}{cccc}\dot{\mathbf{u}}&\dot{\mathbf{z}} \end{array} \right]^{T}=\mathbf{R}\mathbf{i}\mathbf{u}\), in which the matrix \(\mathbf{R}\) is explicitly calculated as \(\mathbf{R}=\left[ \begin{array}{cccc}\mathbf{E}\\ -\mathbf{J}_{z}^{-1}\mathbf{J}_{u}\end{array} \right]\) where \(\mathbf{E}\) is an identity matrix of appropriate dimension, \(\mathbf{J}_{u}=\frac{\partial\mathbf{f}}{\partial\mathbf{u}}\) and \(\mathbf{J}_{z}=\frac{\partial\mathbf{f}}{\partial\mathbf{z}}\), where \(\mathbf{f}(\mathbf{s},t)=0\) is the constraint equation._  
As a consequence of this Lemma, the following important relationships, Equations (14) and (15), can be proved easily. \[\mathbf{R}^{T}\mathbf{J}_{s}^{T}=0 \tag{14}\]  
\[\ddot{\mathbf{s}}=\mathbf{R}\ddot{\mathbf{u}}+\dot{\mathbf{R}}\ddot{\mathbf{u}} \tag{15}\]  
In Equation (14), the Jacobian matrix \(\mathbf{J}_{s}\) can be expressed as follows:  
\[ \begin{array}{ll}\mathbf{J}_{s}&=\frac{\partial\mathbf{f}}{\partial\mathbf{ s}}\\ &=\left[\begin{array}{cccc}\mathbf{\dot{s}}&\partial\mathbf{t}&\frac{ \partial\mathbf{f}}{\partial\mathbf{z}}\end{array}\right]=\left[\begin{array}[] {cccc}\mathbf{J}_{u}&\mathbf{J}_{z}\end{array}\right]\end{array}  \tag{16}\]  
Note that Equations (14) and (15) play a very important role in transforming the complex system Equation (10) into a minimal form. With respect to Equations (14) and (15), multiplying \(\mathbf{R}^{T}\) with both sides of Equation (10) yields  
\[\mathbf{\overline{M}}(\mathbf{s},t)\ddot{\mathbf{u}}+\mathbf{\overline{C}} \big{(}\mathbf{s},\dot{\mathbf{s}},t\big{)}\dot{\mathbf{u}}+\dot{\mathbf{G}}( \mathbf{s},t)\mathbf{u}=\tau_{\mathbf{q}}(t) \tag{17}\]  
where  
\[\mathbf{\overline{M}}=\mathbf{R}^{T}\mathbf{M}(\mathbf{s},t)\mathbf{R}, \mathbf{\overline{C}}=\mathbf{R}^{T}\Big{(}\mathbf{M}(\mathbf{s},t)\dot{ \mathbf{R}}+\mathbf{C}\Big{(}\mathbf{s},\dot{\mathbf{s}},t\big{)}\mathbf{R} \Big{)}\dot{\mathbf{G}}=\mathbf{R}^{T}\mathbf{g}(\mathbf{s},t)\text{, and }\tau_{\mathbf{q}}=\mathbf{R}^{T}\tau(t)\text{.}\]  
Different from the complex ADEs system in Equation (10), which has \(n+2m\) differential equations with the presence of \(2m\) dependent variables \(\mathbf{z}\), and \(2m\) unknown Lagrangian Multipliers \(\lambda\), the minimal form of the dynamic equation, Equation (17), consists of only \(n\) differential equations which are expressed for \(n\) independent generalized coordinates \(\mathbf{u}\) only. Moreover, the matrix form of Equation (17) is similar to the form of any dynamic equation that is usually written for a conventional industrial robot. Hence, by using Equation (17), the dynamic analysis and control law computation for any complex hybrid robot Class III can be implemented in an effective and simplified manner. **Proof of the Lemma. ** Taking a time derivative of the constraint equation \(\mathbf{f}(\mathbf{s},t)=0\) yields  
\[\mathbf{J}_{\mathbf{s}}\dot{\mathbf{s}}=0 \tag{18}\]  
Substituting Equation (16) into Equation (18) obtains  
\[\mathbf{J}_{u}\dot{\mathbf{u}}+\mathbf{J}_{z}\dot{\mathbf{z}}=0 \tag{19}\]  
Rewriting the relationship \(\dot{\mathbf{s}}=\left[ \begin{array}{cc}\dot{\mathbf{u}}&\dot{\mathbf{z}} \end{array}