## 6 Virtual Sensor Design  
The proposed virtual sensor design strategy starts with extracting the lateral forces \(F_{y,f}\) and \(F_{y,r}\) from Equations (6) and (12), which yields  
\[ \begin{split} F_{y,f}&=\frac{1}{\cos(\delta_{f})(L_ {f}+L_{r})}\left(I_{z}\dot{r}+p_{f,f\dot{l}}F_{x,f\dot{l}}+\right.\\ &\left.+p_{f,f\dot{r}}F_{x,f\dot{r}}+p_{f,f\dot{r}}F_{x,r\dot{l}} +p_{f,r\dot{r}}F_{x,r\dot{r}}+\right.\\ &\left.+mL_{r}a_{y}\right),\\ F_{y,r}&=\frac{1}{\cos(\delta_{f})(L_{f}+L_{r})} \left(-I_{z}\dot{r}+p_{r,f\dot{l}}F_{x,f\dot{l}}+\right.\\ &\left.+p_{r,f\dot{r}}F_{x,f\dot{r}}+p_{r,r\dot{l}}F_{x,r\dot{l}} +p_{r,r\dot{r}}F_{x,r\dot{r}}\right.\\ &\left.+mL_{r}a_{y}\right),\end{split}  \tag{14}\]  
where  
\[p_{f,f\dot{l}} =-\sin(\delta_{f})L_{f}-\sin(\delta_{f})L_{r}+L_{a}\cos(\delta_{f }), \tag{16}\] \[p_{f,f\dot{r}} =-\sin(\delta_{f})L_{f}-\sin(\delta_{f})L_{r}-L_{a}\cos(\delta_{f }),\] (17) \[p_{f,r\dot{l}} =-2\sin(\delta_{r})L_{r}+L_{a}\cos(\delta_{r}),\] (18) \[p_{f,r\dot{r}} =-2\sin(\delta_{r})L_{r}-L_{a}\cos(\delta_{r}),\] (19) \[p_{r,f\dot{l}} =L_{a}\cos(\delta_{f}),\] (20) \[p_{r,f\dot{r}} =L_{a}\cos(\delta_{f}),\] (21) \[p_{r,r\dot{l}} =\sin(\delta_{r})L_{f}-\sin(\delta_{r})L_{r}+L_{a}\cos(\delta_{r }),\] (22) \[p_{r,r\dot{r}} =\sin(\delta_{r})L_{f}-\sin(\delta_{r})L_{r}-L_{a}\cos(\delta_{r }). \tag{23}\]  
Bearing in mind the fact that \(F_{x}=ma_{x}\) and then substituting Equations (14) and (15) into Equation (5) yield:  
\[ \begin{split} p_{r}\dot{r}&=F_{x,f\dot{l}}p_{x,f \dot{l}}+F_{x,f\dot{r}}p_{x,f\dot{r}}+\\ &+F_{x,r\dot{l}}p_{x,r\dot{l}}+F_{x,r\dot{r}}p_{x,r\dot{r}}+a_{x}p_ {x}+a_{y}p_{y},\end{split}  \tag{24}\]where  
\[p_{r} =I_{z}\sin(\delta_{f}-\delta_{r}), \tag{25}\] \[p_{x,fl} =-\frac{1}{2}L_{a}\sin(-\delta_{r}+2\delta_{f})+\frac{1}{2}\sin( \delta_{r})L_{a}+\] (26) \[+\cos(\delta_{r})(L_{f}+L_{r}),\] \[p_{x,fr} =\frac{1}{2}L_{a}\sin(-\delta_{r}+2\delta_{f})-\frac{1}{2}\sin( \delta_{r})L_{a}+\] (27) \[+\cos(\delta_{r})(L_{f}+L_{r}),\] \[p_{x,rl} =-\frac{1}{2}L_{a}\sin(\delta_{f}-2\delta_{r})-\frac{1}{2}\sin( \delta_{f})L_{a}+\] (28) \[L_{r} \cos(\delta_{f}-2\delta_{r})+\cos(\delta_{f})L_{f},\] \[p_{x,rr} =\frac{1}{2}L_{a}\sin(\delta_{f}-2\delta_{r})+\frac{1}{2}\sin( \delta_{f})L_{a}+\] (29) \[+L_{r}\cos(\delta_{f}-2\delta_{r})+\cos(\delta_{f})L_{f},\] \[p_{x} =-(L_{f}+L_{r})\cos(\delta_{f})\cos(\delta_{r})m,\] (30) \[p_{x,fl} =-\frac{1}{2}L_{a}\sin(-\delta_{r}+2\delta_{f})+\frac{1}{2}\sin( \delta_{r})L_{a}+\]0  
The state space model of AGV is given by Equations (7)-(10), (14) and (15)  
\[p_{x,fl} =-\frac{1}{2}L_{a}\sin(-\delta_{r}+2\delta_{f})+\frac{1}{2}\sin( \delta_{r})L_{a}+\]1  
where  
\[p_{x,fl} =-\frac{1}{2}L_{a}\sin(-\delta_{r}+2\delta_{f})+\frac{1}{2}\sin( \delta_{r})L_{a}+\]2  
with  
\[p_{x,fl} =-\frac{1}{2}L_{a}\sin(-\delta_{r}+2\delta_{f})+\frac{1}{2}\sin( \delta_{r})L_{a}+\]3  
The unknown input, which has to be estimated by the virtual sensor is given by:  
\[p_{x,fl} =-\frac{1}{2}L_{a}\sin(-\delta_{r}+2\delta_{f})+\frac{1}{2}\sin( \delta_{r})L_{a}+\]4  
Additionally, the system matrices are:  
\[\mathbf{B}(\delta_{f},\delta_{r})= \begin{bmatrix}p_{x}&p_{y}\\ 0&0\\ 0&0\\ 0&0\\ 0&0\end{bmatrix} , \tag{36}\]  
\[\mathbf{E}(\delta_{f},\delta_{r})= \begin{bmatrix}0&-p_{x,fl}&-p_{x,fr}&-p_{x,rl}& -p_{x,rr}\\ -p_{fl}&R_{e}&0&0\\ -p_{r,r}&0&R_{e}&0\\ -p_{r,r}&0&0&0&R_{e}\end{bmatrix} , \tag{37}\]\[\mathbf{G}(\delta_{f},\delta_{r})=\text{diag}(p_{r},1,1,1,1). \tag{38}\]  
Finally, while all state variables \([r,\omega_{fl},\omega_{fr},\omega_{rl},\omega_{rr}]^{T}\) are measured, the output equation is given by  
\[\mathbf{y}=\mathbf{C}\mathbf{x}, \tag{39}\]  
with \(\mathbf{C}=\mathbf{I}\).