## 4 Image-Based Visual Servoing with Limited Field-of-View  
### System Model Development  
After self-calibration of the camera parameters, a new state variable \([\eta_{1},\eta_{2}]^{T}\) is defined as  
\[\eta_{1}=\frac{\bar{u}}{\bar{v}},\eta_{2}=f_{u}\frac{1}{\bar{v}} \tag{12}\]  
to replace the states of (4). Correspondingly, the new desired state variables are defined as  
\[\eta_{1}^{*}=\frac{\bar{u}^{*}}{\bar{v}^{*}},\eta_{2}^{*}=f_{u}\frac{1}{\bar{v }^{*}}.\]  
Taking the time derivative of the new state variables and utilizing (4), the kinematics model is obtained as  
\[\left\{ \begin{array}{l}\dot{\eta}_{1}=-\eta_{2}w\\ \dot{\eta}_{2}=-\frac{\eta_{2}}{Z_{ci}}v+\eta_{1}w=-\frac{f_{u}}{f_{\bar{v}}Y_ {ci}}v+\eta_{1}w.\end{array} \right. \tag{13}\]  
Suppose the two points \(\mathcal{P}_{i}\) and \(\mathcal{P}_{j}\) have the same height in the \(y\) axis direction, i.e., \(Y_{ci}=Y_{cj}\triangleq h>0\), the relationship between the current camera configuration and desired configuration can be deduced through the two points as  
\[\mathcal{P}_{i}^{c}-\mathcal{P}_{j}^{c}=R^{T}\Big{(}\mathcal{P}_{i}^{d}- \mathcal{P}_{j}^{d}\Big{)} \tag{14}\]  
with \(\mathcal{P}^{c}=R^{T}(\mathcal{P}^{d}-t_{r})\) where \(\mathcal{P}^{c},\mathcal{P}^{d}\) denote the points in coordinate of the current camera configuration and the desired camera configuration, respectively. \(R\) and \(t_{r}\) are the rotation matrix and translation vector defined in (3). It is clear that the new state variables have similar geometrical relations with the static points \(\mathcal{P}\). The quantities defined in the image plane \(\eta_{1 ij}=\frac{\bar{u}_{i}}{\bar{v}_{i}}-\frac{\bar{u}_{j}}{\bar{v}_{j}}, \eta_{2 ij}=f_{u}\frac{1}{\bar{v}_{i}}-f_{u}\frac{1}{\bar{v}_{j}}\) have the following relation as  
\[\left[ \begin{array}{cc}\eta_{1 ij}\\ \eta_{2 ij}\end{array} \right]=\left[ \begin{array}{cc}c\theta&-s\theta\\ s\theta&c\theta\end{array} \right]\left[ \begin{array}{cc}\eta_{1 ij}^{*}\\ \eta_{2 ij}^{*}\end{array} \right]. \tag{15}\]  
Then the angle \(\theta\) is recovered by the following equation: \(\sin\theta=(\eta_{1 ij}^{*}\eta_{2 ij}-\eta_{2 ij}^{*}\eta_{1 ij})/(\eta_{1 ij}^{* 2}+\eta_{2 ij}^{*2})\). Therefore, the objective of the visual stabilization task becomes the construction of appropriate velocities \(v,w\) to ensure that  
\[\eta_{1}\rightarrow\eta_{1}^{*},\eta_{2}\rightarrow\eta_{2}^{*},\theta \to 0. \tag{16}\]  
Figure 3: Evolution of the scaled focal length \(f_{u}\). In order to facilitate the subsequent controller design, the error signals are defined as follows  
\[\left[ \begin{array}{c}e_{1}\\ e_{2}\end{array} \right]=\left[ \begin{array}{cc}c\theta&s\theta\\ -s\theta&c\theta\end{array} \right]\left[ \begin{array}{c}\eta_{1}\\ \eta_{2}\end{array} \right]-\left[ \begin{array}{c}\eta_{1}^{*}\\ \eta_{2}^{*}\end{array} \right] \tag{17}\]  
and  
\[e_{0}=\theta. \tag{18}\]  
Considering the time derivative of the error signals and substituting (13) into them, the error system is obtained  
\[\left\{ \begin{array}{c}\dot{e}_{0}=w\\ \dot{e}_{1}=-cvs\theta\\ \dot{e}_{2}=-cvc\theta,\end{array} \right. \tag{19}\]  
where \(c\triangleq\frac{f_{u}}{f\cdot h}>0\) is utilized to simplify the expression of (19). The error system (19) has the similar formulations with the mobile robot kinematics model (1). To overcome the effects of the nonholonomic constraints on the mobile robot visual servoing system, a \(\sigma\)-process is applied to break the one-to-one correspondence of the system. The discontinuous coordinates transformation used [13; 31] is introduced as follows  
\[\left\{ \begin{array}{l}\rho=\sqrt{e_{1}^{2}+e_{2}^{2}}\\ \alpha=atan\Big{(}\frac{e_{1}}{e_{2}}\Big{)}-e_{0}\\ \phi=atan\Big{(}\frac{e_{1}}{e_{2}}\Big{)}\end{array} \right. \tag{20}\]  
for building the system model. The new states in polar coordinates define a diffeomorphism in the region \(\rho\
eq 0\). Then taking the time derivative of these new states, the open-loop system model is developed as  
\[\left\{