## 4 Image-Based Visual Servoing with Limited Field-of-View
### Control Design without Constraints  
According to (17) and (18), the system states in polar coordinate can be obtained directly. There is an unknown constant \(c\) in the system (21), an adaptive controller is designed based on the Lyapunov Theory as following:  
\[v=sign(e_{2})k_{c1}\rho\cos\alpha, \tag{23}\]  
\[w=k_{c2}\alpha+\hat{c}k_{c1}\frac{\sin\alpha\cos\alpha}{\alpha}(\alpha+k_{c3} \phi), \tag{24}\]  
with the adaptive law of \(\hat{c}\) designed as  
\[\dot{\xi}=k_{a}k_{c1}\sin\alpha\cos\alpha(\alpha+k_{c3}\phi), \tag{25}\]where \(k_{c1},k_{c2},k_{c3},k_{a}\in\mathcal{R}^{+}\) are positive control parameters. Then substituting the control inputs (23) and (24) to the open-loop system (21), the closed-loop system can be obtained as  
\[\left\{\begin{array}{ll}\dot{\rho}=-ck_{c1}\rho\cos^{2}\alpha\\ \dot{\alpha}=-k_{c2}\alpha+\bar{c}k_{c1}\sin\alpha\cos\alpha-\bar{c}k_{c1}k_{c3 }\phi\frac{\sin\alpha\cos\alpha}{\alpha}\\ \dot{\phi}=ck_{c1}\sin\alpha\cos\alpha,\end{array}\right. \tag{26}\]  
where \(\bar{c}=c-\bar{c}\) is the parameter estimation error. Note that there exists the limit law \(lim_{a\to 0}\frac{\sin\alpha}{\alpha}=1\), such that there is no singularity problem for the closed-loop system (26) and the control law (24).  
Theorem 1: _The proposed control law (23) and (24) with the parameter adaptive law (25) can drive the system states \(\rho,\alpha,\phi\) to zero in the sense that_  
\[\lim_{t\to\infty}\rho(t)=0,\lim_{t\to\infty}\alpha(t)=0,\lim_{t\to\infty}\phi (t)=0.\]  
Proof of Theorem 1: Choose a Lyapunov function candidate as follows  
\[V=\frac{1}{2}\rho^{2}+\frac{1}{2}\Big{(}\alpha^{2}+k_{c3}\phi^{2}\Big{)}+ \frac{1}{2k_{a}}\bar{c}^{2}. \tag{27}\]  
Taking the time derivative of the function \(V\) and then substituting the closed-loop system (26), we obtain  
\[\dot{V}= \rho\dot{\rho}+\alpha\dot{\alpha}+k_{c3}\phi\phi-\frac{1}{k_{a}} \bar{c}\dot{\bar{c}} \tag{28}\] \[= -ck_{c1}\rho^{2}\cos^{2}\alpha-k_{c2}\alpha^{2}+\bar{c}k_{c1} \alpha\sin\alpha\cos\alpha-\bar{c}k_{c1}k_{c3}\phi\sin\alpha\cos\alpha\] \[+ck_{c1}k_{c3}\phi\sin\alpha\cos\alpha-\bar{c}k_{c1}\sin\alpha \cos\alpha(\alpha+k_{c3}\phi)\] \[= -ck_{c1}\rho^{2}\cos^{2}\alpha-k_{c2}\alpha^{2}.\]  
Since the constant parameter \(c>0\), it is easy to deduce that  
\[\dot{V}\leq 0. \tag{29}\]  
According to (27) and (29), it is easy to concluded that  
\[\rho(t),\alpha(t),\phi(t),\bar{c}(t)\in\mathcal{L}_{\infty} \tag{30}\]  
and to find out the point in the invariant set \(M\)  
\[(\rho=0,\alpha=0)\in M. \tag{31}\]  
Substituting (31) back to the closed-loop system (26) and parameter adaptive law (25), it is concluded that  
\[\hat{c}\phi=0,\dot{\bar{c}}=0. \tag{32}\]  
From (30), we know that \(\hat{c}\in\mathcal{L}_{\infty}\). It is concluded that \(\hat{c}\) is a nonzero constant. Thus  
\[\phi=0. \tag{33}\]  
According to above analysis, it is obvious that the invariant set \(M\) only consists of the equilibrium point being \((\rho=0,\alpha=0,\phi=0,\hat{c}=constant)\in M\). Based on [PERSON]'s invariance principle [32], we can infer that the system states converge to the equilibrium point asymptotically. The Theorem 1 is proved.  
When the states of the closed-loop system converge to zero asymptotically, a singularity problem will happen. To deal with this problem that the state \(\rho\) go to zero, an intuitive method is to set the linear velocity \(v\) to zero when the state \(\rho\) is smaller than a threshold. The angular velocity is designed as a pure proportional controller \(w=-k_{w}\theta\), with \(k_{w}\) being a positive control gain. This is scheduled as the last stage.