## 3 Mobile Robot Modeling
### Dynamics  
Within the Newton-Euler formulation applied to multibody systems, we distinguish three rigid bodies composing the DWMR, as shown in Figure 5. The \(6\times 6\)_intertia dyad_ of the two actuated wheels are denoted by \(\mathbf{M}_{1}\) and \(\mathbf{M}_{2}\), with a similar notation for their six-dimensional twists. Since the platform undergoes planar motion, its \(3\times 3\) counterpart inertia dyad \(\mathbf{M}_{3}^{\prime}\) is introduced here, while the platform twist becomes, correspondingly, the three-dimensional vector \(\mathbf{t}_{3}^{\prime}\). Besides, the transformation matrices of the three moving bodies are expressed in a similar way. These relate the body twists with the vector of actuated joint rates \(\hat{\mathbf{\theta}}\), introduced in Equation (6), i.e.,  
\[\mathbf{t}_{i}=\mathbf{T}_{i}\hat{\mathbf{\theta}},\quad i=1,2 \tag{11}\]  
and  
\[\mathbf{t}_{3}^{\prime}=\mathbf{T}_{3}^{\prime}\hat{\mathbf{\theta}} \tag{12}\]  
where, from Equations (1), (5) and (8), and considering that the two wheels undergo general six-degree-of -freedom motion,  
\[\mathbf{T}_{1} = \begin{bmatrix}-\mathbf{i}-r/(2l)\mathbf{k}&r/(2l)\mathbf{k}\\ r\mathbf{j}&\mathbf{0}\end{bmatrix} ,\] \[\mathbf{T}_{2} = \begin{bmatrix}-r/(2l)\mathbf{k}&-\mathbf{i}+r/(2l)\mathbf{k} \\ \mathbf{0}&r\mathbf{j}\end{bmatrix} , \tag{13}\] \[\mathbf{T}_{3}^{\prime} =\frac{r}{2} \begin{bmatrix}-1/l&1/l\\ (a/l)\mathbf{i}+\mathbf{j}&(-a/l)\mathbf{i}+\mathbf{j}\end{bmatrix} \]  
By means of the natural orthogonal complement [34], the generalized dynamics model is derived from the Newton-Euler equations for the DWMR, which leads to  
\[\mathbf{I}\ddot{\mathbf{\theta}}=-\mathbf{C}\dot{\mathbf{\theta}}+\mathbf{\tau}+\delta+ \mathbf{\gamma} \tag{14}\]  
where \(\mathbf{I}\) is the positive definite \(2\times 2\) generalized inertia matrix, \(\mathbf{C}\dot{\mathbf{\theta}}\) being the 2-dimensional vector of Coriolis and centrifugal-force terms. Furthermore, \(\mathbf{\tau}\), \(\delta\) and \(\mathbf{\gamma}\) denote the 2-dimensional vectors of generalized active, dissipative, and gravity forces, respectively. Moreover, the twist \(\mathbf{t}\) is a linear transformation of the independent generalized speeds \(\dot{\mathbf{\theta}}\), as per Equation (8). The \(2\times 2\) generalized inertia matrix \(\mathbf{I}\) is obtained as  
\[\mathbf{I}=\mathbf{I}_{w}+\mathbf{I}_{p}=\sum_{i=1}^{2}\mathbf{T}_{i}^{T} \mathbf{M}_{i}\mathbf{T}_{i}+(\mathbf{T}_{3}^{\prime})^{T}\mathbf{M}_{3}^{ \prime}\mathbf{T}_{3}^{\prime} \tag{15}\]  
where \(\mathbf{I}_{w}\) and \(\mathbf{I}_{p}\) are the inertia matrices of the two actuated wheels and of the platform, respectively. The \(2\times 2\) matrix \(\mathbf{C}\) of Coriolis and centrifugal force is derived below:  
\[\mathbf{C}=\sum_{i=1}^{2}\mathbf{T}_{i}^{T}\mathbf{M}_{i}\mathbf{T}_{i}+( \mathbf{T}_{3}^{\prime})^{T}\mathbf{M}_{3}^{\prime}\mathbf{T}_{3}^{\prime}+ \sum_{i=1}^{2}\mathbf{T}_{i}^{T}\mathbf{W}_{i}\mathbf{M}_{i}\mathbf{T}_{i}+( \mathbf{T}_{3}^{\prime})^{T}\mathbf{W}_{3}^{\prime}\mathbf{M}_{3}^{\prime} \mathbf{T}_{3}^{\prime} \tag{16}\]  
where \(\mathbf{W}_{i}\), for \(i=1,2\), and \(\mathbf{W}_{3}^{\prime}\) are, correspondingly, \(6\times 6\) and \(3\times 3\)_angular-velocity dyads_, defined as  
\[\mathbf{M}_{i}\equiv \begin{bmatrix}\mathbf{I}_{i}&\mathbf{O}_{3}\\ \mathbf{O}_{3}&m_{i}\mathbf{I}_{3}\end{bmatrix} ,\quad\mathbf{W}_{i}\equiv \begin{bmatrix}\mathbf{\Omega}_{i}&\mathbf{O}_{3}\\ \mathbf{O}_{3}&\mathbf{O}_{3}\end{bmatrix} ,\quad\mathbf{W}_{3}^{\prime}\equiv \begin{bmatrix}\omega_{3}&\mathbf{0}_{2}^{T}\\ \mathbf{0}_{2}&\mathbf{O}_{2}\end{bmatrix}