# Trajectory Tracking Control Method Based on Adaptive Higher Order Sliding Mode
## 2 Design of Adaptive High-Order Sliding Mode Trajectory Tracking Controller  
### Establishing Trajectory Tracking Control Model  
The kinematic and dynamic models of the missile based on the geodetic coordinate system in the two-dimensional plane are constructed as follows [17; 18]:  
\[\left\{\begin{array}{l}\dot{x}=v\cos\theta\\ \dot{y}=v\sin\theta\\ \ddot{v}=g(n_{x}-\sin\theta)\\ \dot{\theta}=\frac{\kappa}{v}(n_{y}-\cos\theta)\end{array}\right. \tag{1}\]  
where \(v\) is the flight speed of the missile, \(\theta\) is the trajectory inclination angle, \(x\) is the longitudinal position coordinate, and \(y\) is the height position coordinate, \(\dot{v}\), \(\dot{\theta}\), \(\dot{x}\) and \(\dot{y}\) are the derivatives of \(v\), \(\theta\), \(x\), and \(y\), respectively, while \(n_{x}\) and \(n_{y}\) are the overload values in the \(x\) and \(y\) directions, respectively. The acceleration of gravity is \(g\) and is equal to 9.81 m/s\({}^{2}\). \(n_{x}\) is expressed as  
\[n_{x}=\frac{X}{mg} \tag{2}\]  
where \(m=1030\) kg is the mass of the missile, and resistance \(X\) is expressed as  
\[X=\frac{1}{2}\rho v^{2}\mathrm{C}_{x}S \tag{3}\]where \(C_{x}=0.1\), \(S=0.5\) m\({}^{2}\), and the density \(\rho\) is expressed as  
\[\rho=1.225e^{-\frac{y}{7110}} \tag{4}\]  
According to the modeling methods reported in [18], the equation of motion of a particle and the model of the error tracking controller are obtained as follows:  
\[\left(\begin{array}{c}\dot{y}\\ \dot{\theta}\end{array}\right)=\left(\begin{array}{c}v\sin\theta\\ -\frac{g\cos\theta}{\rho}\end{array}\right)+\left(\begin{array}{c}0\\ \frac{g}{\rho}\end{array}\right)n_{y} \tag{5}\]  
\[x=\left(\begin{array}{c}x_{1}\\ x_{2}\end{array}\right)=\left(\begin{array}{c}y-y_{n}\\ \theta-\theta_{n}\end{array}\right) \tag{6}\]  
where state with subscript \(n\) represents the nominal state value.  
\[\begin{array}{rl}\dot{x}&=\left(\begin{array}{c}\dot{x}_{1}\\ \dot{x}_{2}\end{array}\right)=\left(\begin{array}{c}\dot{y}-\dot{y}_{n}\\ \theta-\dot{\theta}_{n}\end{array}\right)\\ &=\left(\begin{array}{c}v_{n}\sin\theta-v_{n}\sin\theta_{n}\\ \frac{g}{\rho_{n}}n_{y}-\frac{g}{\rho_{n}}\cos\theta-\frac{g}{\rho_{n}}n_{y_{ n}}+\frac{g}{\rho_{n}}\cos\theta_{n}\\ &=\left(\begin{array}{c}v_{n}\sin\theta-v_{n}\sin\theta_{n}\\ -\frac{g}{\rho_{n}}\cos\theta+\frac{g}{\rho_{n}}\cos\theta_{n}\end{array} \right)+\left(\begin{array}{c}0\\ \frac{g}{\rho_{n}}\end{array}\right)\Delta n_{y}\end{array} \tag{7}\]  
where the uncontrolled variable \(v\) uses the state value on the nominal trajectory at the corresponding moment and is marked with subscript \(n\). For the overload deviation, \(\Delta n_{y}=n_{y}-n_{y_{n}}\) is taken as the control variable. Let \(u_{y}=\Delta n_{y}\), then the above model can be transformed into  
\[\left\{\begin{array}{l}\dot{x}_{1}=V_{n}\sin\theta-V_{n}\sin\theta_{n}\\ \dot{x}_{2}=-\frac{g}{V_{n}}\cos\theta+\frac{g}{V_{n}}\cos\theta_{n}+\frac{g }{V_{n}}u_{y}\end{array}\right. \tag{8}\]  
At this point, the mid-guidance for trajectory tracking control model based on the nominal trajectory has been established.