## 5 Mathematical Model of the Automated Guided Vehicle  
Before proceeding with the development, let us provide the names of the crucial variables shaping the AGV dynamics, which are given in Table 2.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **Unit** & **Value** \\ \hline \(m\) & kg & 50 \\ \(L_{f}\) & m & 0.25 \\ \(L_{r}\) & m & 0.25 \\ \(l_{z}\) & kg/m\({}^{2}\) & 89.18 \\ \(C_{r}\) & kg/\({}^{\circ}\) & 1.86 \\ \(C_{f}\) & kg/\({}^{\circ}\) & 1.86 \\ \(L_{a}\) & m & 0.16 \\ \(R_{e}\) & m & 0.09 \\ \(I_{xw}\) & kg/m\({}^{3}\) & 0.00209089 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Automated Guided Vehicles (AGVs) parameters.  
Figure 3: Suspension system.  
From Figure 2, it can be derived that the force causing longitudinal motion is given by  
\[F_{x} =\cos(\delta_{f})(F_{x,fl}+F_{x,fr})+\cos(\delta_{r})(F_{x,rl}+F_{x, rr})+\] \[-\sin(\delta_{f})F_{y,f}-\sin(\delta_{r})F_{y,r}. \tag{5}\]  
Analogously, the lateral forces can be analyzed:  
\[F_{y} =\sin(\delta_{f})(F_{x,fl}+F_{x,fr})+\sin(\delta_{r})(F_{x,rl}+F_ {x,rr})+\] \[+\cos(\delta_{f})F_{y,f}+\cos(\delta_{r})F_{y,r}, \tag{6}\]  
where the longitudinal wheel forces obey:  
\[I_{xw}\dot{\omega}_{fl} =p_{f,l}T-F_{x,fl}R_{e}, \tag{7}\] \[I_{xw}\dot{\omega}_{fr} =p_{f,r}T-F_{x,fr}R_{e},\] (8) \[I_{xw}\dot{\omega}_{rl} =p_{r,l}T-F_{x,rl}R_{e},\] (9) \[I_{xw}\dot{\omega}_{rr} =p_{r,r}T-F_{x,rr}R_{e}, \tag{10}\]  
with the torque distribution coefficients \(p_{i,j}\geq 0\) satisfying:  
\[p_{f,l}+p_{f,r}+p_{r,l}+p_{r,r}=1. \tag{11}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline \(\mathbf{COG}\) & **Center of Gravity** \\ \hline \(i=front,rear\) & axle location \\ \(j=left,rightright\) & wheel location \\ \(L_{f}\) & distance between front axle and COG \\ \(L_{r}\) & distance between rear axle and COG \\ \(L_{a}\) & rear/front half gauge \\ \(r\) & yaw rate \\ \(v_{x}\) & longitudinal velocity \\ \(a_{x}\) & longitudinal acceleration \\ \(a_{y}\) & lateral acceleration \\ \(m\) & mass \\ \(\delta_{f}\) & steering angle of the front wheels \\ \(\delta_{r}\) & steering angle of the rear wheels \\ \(F_{x}\) & sum of forces causing longitudinal motion \\ \(F_{y}\) & sum of forces causing lateral motion \\ \(F_{x,ij}\) & longitudinal force on \(i,j\) wheel \\ \(F_{y,f}\) & total lateral force on the front wheels \\ \(F_{y,r}\) & total lateral force on the rear wheels \\ \(\omega_{i,j}\) & angular velocity of \(i,j\) wheel \\ \(T\) & total torque acting on all wheels \\ \(p_{i,j}\) & torque distribution coefficient \\ \(I_{xw}\) & wheel moment of inertia \\ \(I_{z}\) & robot moment of inertia around \(z\)-axis \\ \(R_{e}\) & wheel effective radius \\ \(C_{f}\) & front wheel cornering stiffness \\ \(C_{r}\) & rear wheel cornering stiffness \\ \hline \hline \end{tabular}
\end{table}
Table 2: Notation.  
It should also be noticed that \(p_{i,j}\) are assumed to be known parameters. Finally, the yaw rate dynamics is given by  
\[\begin{split} I_{z}\dot{r}&=L_{f}(\sin(\delta_{f})(F_{ x,f\dot{l}}+F_{x,f\dot{r}})+\cos(\delta_{f})F_{y,f})+\\ &+L_{a}\cos(\delta_{f})(F_{x,f\dot{r}}-F_{x,f\dot{l}})+L_{r}(\sin (\delta_{r})(F_{x,r\dot{l}}+F_{x,r\dot{r}})\\ &-\cos(\delta_{r})F_{y,r})+L_{a}\cos(\delta_{r})(F_{x,r\dot{r}}- F_{x,r\dot{l}}).\end{split} \tag{12}\]  
Having a mathematical description of AGV, the objective of the subsequent part of this paper is to develop a set of virtual sensors enabling estimation of \(F_{x,f\dot{r}}\), \(F_{x,f\dot{l}}\), \(F_{x,r\dot{r}}\), \(F_{x,r\dot{l}}\) and \(T\) based on the available measurement vector:  
\[y=[r,\omega_{f\dot{l}},\omega_{f\dot{r}},\omega_{r\dot{l}},\omega_{rr}]^{T}, \tag{13}\]  
as well as lateral and longitudinal accelerations \(a_{x}\) and \(a_{y}\) and known steering angles \(\delta_{f}\) and \(\delta_{r}\) being available control variables.