# BIZON-UGV for Airport Pavement Testing: Mechanics and Control
## 3 Modelling and Control
### Mathematical Model  
As shown in Figure 7, the model of the Bizon platform can be described considering generalized coordinates \(\mathbf{q}(t)= \begin{bmatrix}x_{c}&y_{c}&\Theta_{r}&\Theta_{l}\end{bmatrix} ^{T}\) in global reference frame XY. The body frame origin lies at the center of mass (CoM), where a pair \((x_{c},y_{c})\) describes the CoM position, \((\Theta_{r},\Theta_{l})\) are angles of the right and left wheels, and \(\psi\) is the platform heading angle. As described in [15], the mobile platform dynamics can be represented by the following motion equation considering friction, Coriolis, and centrifugal forces, but also non-holonomic constraints:  
\[\mathbf{M}(\mathbf{q})\ddot{\mathbf{q}}+\mathbf{B}\ddot{\mathbf{q}}+\mathbf{C} \big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}+\mathbf{A}^{T}(\mathbf{q})\lambda(t )=\mathbf{E}\tau(t). \tag{7}\]  
Figure 6: Deformation results, actuator beam, and fork of wheel system. Figure 7: Mobile platform model. Differential wheel approach. In Equation (7), \(\mathbf{M}(\mathbf{q})\) is a mass and inertia matrix, and \(\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\) is a Coriolis and centrifugal vector:  
\[\mathbf{M}(\mathbf{q})= \begin{bmatrix}m_{b}+2m_{wt}&0&-\frac{m_{wt}rd}{b}s \psi&\frac{m_{wt}rd}{b}s\psi\\ 0&m_{b}+2m_{wt}&\frac{m_{wt}rd}{b}c\psi&-\frac{m_{wt}rd}{b}c\psi\\ -\frac{m_{wt}rd}{b}s\psi&\frac{m_{wt}rd}{b}c\psi&m_{33}&m_{34}\\ -\frac{m_{wt}rd}{b}s\psi&-\frac{m_{wt}rd}{b}c\psi&m_{43}&m_{44}\\ \end{bmatrix} ,\ \mathbf{C}(\mathbf{q},\dot{\mathbf{q}})= \begin{bmatrix}-2m_{wt} \dot{\psi}^{2}c\psi\\ -2m_{wt}\dot{\psi}^{2}s\psi\\ 0\\ 0\\ \end{bmatrix} ,\]  
where \(m_{b}\) is the mass of the platform base, \(m_{wt}\) is the mass of the wheels with the track, \(r\) is the substitute wheel radius, \(I_{zzb}\) is the base moment of inertia, \(I_{yyw}\) and \(I_{zzw}\) are moments related to the wheels, \(b\) is the platform width, and \(d\) is the distance between the center of mass and wheel axis. Mass coefficients \(m_{33}\), \(m_{34}\), \(m_{43}\), \(m_{44}\) are defined as follows:  
\(m_{33}=r^{2}\big{(}2m_{w}\big{[}b^{2}+d^{2}\big{]}+I_{zzb}+2I_{zzw}\big{)}/(4b ^{2})+I_{yyw}\),  
\(m_{34}=-r^{2}\big{(}2m_{w}\big{[}b^{2}+d^{2}\big{]}+I_{zzb}+2I_{zzw}\big{)}/(4b ^{2})\),  
\(m_{43}=m_{34}\),  
\(m_{44}=r^{2}\big{(}2m_{w}\big{[}b^{2}+d^{2}\big{]}+I_{zzb}+2I_{zzw}\big{)}/(4b ^{2})+I_{yyw}\). The other matrices in (7) denote the following: \(\mathbf{A}(\mathbf{q})\) is a constraint matrix, \(\mathbf{B}\) is a friction coefficient matrix, and \(\mathbf{E}\) is a matrix related to the wheel torques:  
\[\mathbf{A}(\mathbf{q})= \begin{bmatrix}-s\psi&c\psi&0&0\\ -c\psi&-s\psi&\frac{r}{2}&\frac{r}{2}\end{bmatrix} ,\ \mathbf{E}= \begin{bmatrix}0&0\\ 0&0\\ 1&0\\ 0&1\end{bmatrix} ,\ \mathbf{B}= \begin{bmatrix}b_{11}&0&0&0\\ 0&b_{22}&0&0\\ 0&0&b_{33}&0\\ 0&0&b_{44}\end{bmatrix} ,\]  
where \(b_{11}\), \(b_{22}\) are coefficients related to the linear motion and \(b_{33}\), \(b_{44}\) are coefficients related to angular speed. Matrix \(\mathbf{S}(\mathbf{q})\) is introduced to omit constraints of the mobile vehicle and performs \(\mathbf{A}(\mathbf{q})\mathbf{S}(\mathbf{q})=0\). The \(\mathbf{S}(\mathbf{q})\) matrix has the following form:  
\[\mathbf{S}(\mathbf{q})= \begin{bmatrix}\frac{r}{2}c\psi&\frac{r}{2}c\psi\\ \frac{r}{2}s\psi&\frac{r}{2}s\psi\\ 1&0\\ 0&1\end{bmatrix} ,\ \text{where}\ s\psi=sin\psi\text{and}\ c\psi=cos\psi.\]  
Multiplying Equation (7) by \(\mathbf{S}^{T}(\mathbf{q})\)  
\[\mathbf{S}^{T}(\mathbf{q})\mathbf{M}(\mathbf{q})\mathbf{\ddot{q}}+\mathbf{S}^ {T}(\mathbf{q})\mathbf{B}\mathbf{\ddot{q}}+\mathbf{S}^{T}(\mathbf{q})\mathbf{ C}(\mathbf{q},\mathbf{\ddot{q}})=\boldsymbol{\tau}(t) \tag{8}\]  
and introducing wheel speed \(\mathbf{v}= \begin{bmatrix}\dot{\Theta}_{r}&\dot{\Theta}_{l}\end{bmatrix}