## 4 The Reachable Zone Predictive Method
### Reachable Zone Determination Method  
In this subsection, a six-point method (SPM) is proposed to determine the reachable zone of a gliding vehicle at a certain moment. The SPM focuses on quick discrimination of the vehicle coverage capability, providing a discriminatory condition for the online target adjustment and replanning on whether a target point is reachable or not. It is noteworthy that this method provides the foundation for the design of the database method that follows later.  
Once the profile for the maximum and minimum range has been designed, the longitudinal movement of the vehicle is determined immediately. As shown in Figure 3, the bank-less reversal procedure can be used to immediately find the points \"1\", \"3\", \"4\", and \"6\" with little additional effort. Two unreachable virtual target points are artificially selected on the initial heading; by tracking these two points it is possible to find points \"2\" and \"5\" in the D-V profile tracking method.  
The SPM procedure of the reachable zone corresponding to the different moments of the vehicle is as follows:  
**Step 1:** Calculate the position for the minimum range point \"2\" and the maximum range point \"5\" by tracking the designed profile as shown in Figure 1;  
**Step 2:** Calculate the maximum cross-range points \"1\" and \"3\" under the minimum-range condition by keeping the bank angle positive or negative during the whole gliding phase and employ the same method to calculate the maximum cross-range points \"4\" and \"6\" under the maximum-range condition;  
**Step 3:** Connect the six predictive points by the simple interpolation method, to form a sector-shaped reachable zone in the longitude-latitude plane as shown in Figure 4. When six points are known, an online quick determination of the reachable area can be performed. Therefore, the six points, the area, the downrange, and the cross-range are used to describe the properties of the reachable zone;  
Figure 3: The six boundary points’ schematic.  
**Step 4:** Determine the reachability of a new target point when one is given. As shown in Figure 4, a new target point (\(\lambda_{t}\), \(\phi_{t}\)) is randomly placed in a zone. Establish the minimum range reachability curve \(l_{1}\), the maximum range reachability curve \(l_{2}\), and the lateral boundary lines \(l_{3}\) and \(l_{4}\). The mathematical expressions are as follows:  
\[\begin{split} l_{1}:&\ x=a_{1}y^{2}+b_{1}y+c_{1}\\ l_{2}:&\ x=a_{2}y^{2}+b_{2}y+c_{2}\\ l_{3}:&\ y=\frac{\phi_{4}-\phi_{1}}{\lambda_{4}- \lambda_{1}}(x-\lambda_{1})+\phi_{1}\\ l_{4}:&\ y=\frac{\phi_{4}-\phi_{1}}{\lambda_{6}- \lambda_{3}}(x-\lambda_{3})+\phi_{3}\end{split} \tag{17}\]  
where \(a_{i}\), \(b_{i}\), and \(c_{i}\), with \(i=1\) or \(2\), can be easily obtained by solving the quadratic expressions for the six boundary points with simple algebraic skills, so they are omitted here.  
If the new point's longitude and latitude satisfy the conditions as follows:  
\[\begin{split}\lambda_{t}>a_{1}\phi_{t}{}^{2}+b_{1}\phi_{t}+c_{1} \\ \lambda_{t}<a_{2}\phi_{t}{}^{2}+b_{2}\phi_{t}+c_{2}\\ \phi_{t}<\frac{\phi_{4}-\phi_{1}}{\lambda_{4}-\lambda_{1}}( \lambda_{t}-\lambda_{1})+\phi_{1}\\ \phi_{t}>\frac{\phi_{6}-\phi_{3}}{\lambda_{6}-\lambda_{3}}( \lambda_{t}-\lambda_{3})+\phi_{3}\end{split} \tag{18}\]  
then the new target point is reachable; otherwise, it is unreachable.