# Curved Path Following Control for a Small Fixed-Wing UAV with Parameters Adaptation
## 3 Controller Design
### Kinematic Controller Design with Fixed Control Parameters  
**Theorem 1**.: _Considering the kinematic error model of the UAV described in Equation (7), the following control law \(u_{cmd}\),_  
\[u_{cmd}=-k_{1}V\|\
abla f\|f_{sat}(e_{d})+\frac{d}{dt}tan^{-1}\left(-\frac{f _{x}}{f_{y}}\right)-k_{2}V^{2}\|\
abla f\|\sin(e_{\chi}) \tag{8}\]  
Figure 2: The sketch of the solution to the curved-path following controller with parameters adaptation.  
_where \(k_{1}\) and \(k_{2}\) are all positive constants, \(f_{sat}(x)\) is the saturation function defined as_  
\[f_{sat}(x)=\left\{\begin{array}{rl}x_{0},&x>x_{0}\\ x,-x_{0}\leq x\leq x_{0}\\ -x_{0},&x<x_{0}\end{array}\right.\]  
_with \(x_{0}\) an arbitrary given positive constant, asymptotically drives \(e_{d}\) and \(e_{\chi}\) towards zero._  
**Proof of Theorem 1.** Substitute \(u_{cmd}\) shown as Equation (8) into Equation (7), then  
\[\left\{\begin{array}{rl}\dot{e}_{d}&=V\|\
abla f\|\sin(e_{\chi})\\ \dot{e}_{\chi}&=-k_{1}V\|\
abla f\|f_{sat}(e_{d})-k_{2}V^{2}\|\
abla f\|\sin(e _{\chi})\\ &=-k_{1}V\|\
abla f\|f_{sat}(e_{d})-k_{2}V\dot{e}_{d}\end{array}\right. \tag{9}\]  
Define the domain \(D_{1}=\{(e_{d},e_{\chi})||e_{d}|<e_{d1}|e_{\chi}|<\pi\}\), where \(e_{d1}\) is a positive constant, and let \(x=[e_{d},e_{\chi}]^{T}\), Equation (9) can be rewritten as  
\[\dot{x}=g(\mathbf{x},t) \tag{10}\]  
where  
\[g(\mathbf{x},t)=\left[\begin{array}{rl}V\|\
abla f\|\sin(e_{\chi})\\ -k_{1}V\|\
abla f\|f_{sat}(e_{d})-k_{2}V^{2}\|\
abla f\|\sin(e_{\chi})\end{array}\right] \tag{11}\]  
It can be found that \(g:\,D_{1}\to R^{2}\) is a locally Lipschitz map from the domain \(D_{1}\subset R^{2}\) into \(R^{2}\) with **Assumptions 1** and 2 adopted. \([0,\,0]^{T}\) is the only equilibrium point for (11) in \(D_{1}\). Let \(U(\mathbf{x})\) be a candidate Lyapunov function given as:  
\[U(\mathbf{x})=k_{1}\int_{0}^{e_{d}}f_{sat}(y)dy+(1-cos(e_{\chi})). \tag{12}\]  
Therefore, \(U(\mathbf{0})=0\). Moreover, if \(e_{d}\
eq 0\,k_{1}\int_{0}^{e_{d}}f_{sat}(y)dy>0\). If \(e_{\chi}\
eq 0,(1-cos(e_{\chi}))>0\); thus \(U(\mathbf{x})>0\) in \(D_{1}-\{\mathbf{0}\}\). The time derivative of \(U(\mathbf{x})\) along the trajectory of (9) is  
\[\begin{array}{rl}\dot{U}(\mathbf{x})&=k_{1}\dot{e}_{d}f_{\rm sat}(e_{d})+\dot{e} _{\chi}\sin(e_{\chi})\\ &=k_{1}V\|\
abla f\|f_{\rm sat}(e_{d})\sin(e_{\chi})\\ &+\big{(}-k_{1}\|\
abla f\|Vf_{\rm sat}(e_{d})-k_{2}V^{2}\|\
abla f\|\sin(e_{ \chi})\big{)}\sin(e_{\chi})\\ &=-k_{2}V^{2}\|\
abla f\|\sin^{2}(e_{\chi})\end{array} \tag{13}\]  
It can be found that \(\dot{U}(\mathbf{x})\) is negative semidefinite. Since \(V\), \(f_{x}\), \(f_{y}\), \(f_{xx}\), \(f_{xy}\), and \(f_{yy}\) are bounded and \(\|\
abla f\|\geq\lambda\), one can derive that \(d\|\
abla f\|/dt\) is bounded. It can be found from (12) that \(U(\mathbf{x})\) is lower bounded. Since \(V\), \(\|\
abla f\|\), and their derivatives are all bounded, by (13) \(\dot{U}(\mathbf{x})\) is negative semidefinite and uniformly continuous in time. By [PERSON]'s lemma [31], \(\dot{U}(\mathbf{x})\to 0\) as \(t\to\infty\). Moreover, it has \(sin(e_{\chi})\to 0\), and \(\dot{e}_{d}\to 0\) as in (11). Since \(e_{d}\) is bounded, it follows that \(e_{d}\) tends to a finite limit \(\overline{e}_{d}\) as \(t\to\infty\). Since \(e_{\chi}\to 0\) as \(t\to\infty\), and \(\dot{e}_{\chi}\) in (11) is uniformly continuous, it derives that \(\dot{e}_{\chi}\to 0\) as \(t\to\infty\). Hence, it follows that \(\lim\limits_{t\to\infty}f_{sat}(e_{d})\to 0\), and \(\overline{e}_{d}=0\). Therefore, the kinematic control law (8) can asymptotically drive \(e_{d}\) and \(e_{\chi}\) toward zero. \(\square\)