# Multivariable Adaptive Super-Twisting Guidance Law Based on Barrier Function
## 2 Problem Statement
### Kinematics of Missile-Target Engagement  
Figure 1 demonstrates the three-dimensional pursuit geometry relationship between missile and target, where \(\left[ \begin{array}{cc}\mathbf{e}_{r}&\mathbf{e}_{\theta}&\mathbf{e}_{\phi}\end{array}  \right]^{T}\) is a unit vector aligned with the three axes of the LOS coordinate system. Denoted \((X,Y,Z)\) and \((r,\theta,\phi)\) as the inertial coordinate system and the line-of-sight (LOS) coordinate system, respectively, where \(r\), \(\theta\) and \(\phi\) represent the relative distance between the point masses of missile and target, the azimuth and elevation angles of the LOS, respectively. Then, the missile-target engagement kinematics can be described by the LOS coordinate system with the origin at the mass center of the missile, and the relative velocity in LOS coordinate system \(\mathbf{V}\) is given by:  
\[\mathbf{V}=\left[ \begin{array}{cc}\dot{r}&\dot{r}\dot{\theta}\cos\phi&r\dot{ \phi}\end{array} \right]^{T} \tag{1}\]  
where \(\dot{\theta}\) and \(\dot{\phi}\) represent the azimuth and elevation angular rates of LOS, respectively. The relative velocity satisfies the following vector differential form:  
\[\frac{d\mathbf{V}}{dt}=\frac{\delta\mathbf{V}}{\delta t}+\mathbf{\omega}\times\mathbf{V}=\mathbf{ a}_{T}-\mathbf{a}_{M} \tag{2}\]  
where \(d\mathbf{V}/dt\) and \(\delta\mathbf{V}/\delta t\) denote the absolute derivative of \(\mathbf{V}\) in an inertial coordinate system, and the relative derivative of \(\mathbf{V}\) in an LOS coordinate system, respectively; \(\mathbf{a}_{T}\) and \(\mathbf{a}_{M}\) denote the accelerations of the target and missile, respectively; \(\mathbf{\omega}\) denotes the relative rotation angular velocity of LOS coordinate system in inertial coordinate, which satisfies the following expression:  
\[\mathbf{\omega}=\left[ \begin{array}{cc}\dot{\theta}\sin\phi&-\dot{\phi}&\dot{ \theta}\cos\phi\end{array} \right]^{T} \tag{3}\]  
Substituting Equations (1) and (3) into Equation (2), the three-dimension missile-target engagement kinematic equations can be formulated as:  
\[\left\{ \begin{array}{l}\ddot{r}=\dot{\theta}^{2}\cos^{2}\phi r+\dot{\phi}^ {2}r+a_{T}-a_{Mr}\\ \ddot{\theta}=2\dot{\phi}\dot{\theta}\tan\phi-\frac{2\dot{r}\dot{\theta}}{r}+ \frac{a_{T\theta}}{r\cos\phi}-\frac{a_{M\theta}}{r\cos\phi}\\ \ddot{\phi}=-\dot{\theta}^{2}\sin\phi\cos\phi-\frac{2\dot{r}\dot{\theta}}{r}+ \frac{a_{T\theta}}{r}-\frac{a_{M\phi}}{r}\end{array} \right. \tag{4}\]  
Figure 1: Three-dimension schematic diagram of missile-target engagement. where \(\left[ \begin{array}{cc}a_{T\gamma}&a_{T\theta}&a_{T\phi}\end{array} \right]^{T}\) and \(\left[ \begin{array}{cc}a_{M\gamma}&a_{M\theta}&a_{M\phi}\end{array} \right]^{T}\) are components of target and missile accelerations along the LOS coordinate axes, respectively. **Remark 1**. : _From Equation (4), one can easily conclude that \(r=0\), and \(\phi=\pm\pi/2\) are singular points of the control system. However, in real practice, the distance between point masses of missile and target tends to be zero but not zero due to their realistic physical structure. Thus, the scenario \(r=0\) does not occur throughout the guidance process. As for regular points \(\phi=\pm\pi/2\), the study work in [14] in the planar homing engagement phase proved that these points are not the stable equilibrium cases. Hence, the guidance process just crosses them without staying._  
In consideration of the constraint of desired LOS terminal angles \(\theta_{d}\) and \(\phi_{d}\), denote \(\mathbf{x}_{1}=\left[ \begin{array}{cc}x_{1\theta}&x_{1\phi}\end{array} \right]^{T} =\left[ \begin{array}{cc}\theta-\theta_{d}&\phi-\phi_{d}\end{array} \right]^{T}\) and \(\mathbf{x}_{2}=\dot{x}_{1}=\left[ \begin{array}{cc}x_{2\theta}&x_{2\phi}\end{array}  \right]^{T}=\left[ \begin{array}{cc}\dot{\theta}&\dot{\phi}\end{array} \right]^ {T}\) as the control variables, and substitute into Equation (4) yields:  
\[