# Hyperbolic Asynchronous Method of a Radio Navigation Technique
## 2 Basic of the Hyperbolic Technique
### Selected Localization Algorithms for Hyperbolic Solution  
On the basis of the literature studies and previous professional practice, and taking into account the adopted system features of the asynchronous navigation system being developed, for the purpose of solving Equation (4), the usefulness of the following locating algorithms was compared:  
* Chan algorithm [19],
* [PERSON]'s algorithm [20],
* Fang algorithm [21],  
whose names result from the names of the authors of these algorithms. However, due to the limitation of the Fang algorithm, which can only be used for a maximum of three reference stations, the performance of the above algorithms has been tested for such a case. However, this does not limit the conclusions of such an analysis, which also apply to a larger number of reference stations. The first one, the Chan algorithm, is one of the non-iterational methods of high utility importance, which provides satisfactory results regardless of the distance from the localized terminal to the reference stations. The basis for the operation of this algorithm is the following matrix, dependent on the user terminal coordinates \((x_{UIT},y_{UIT})\)[19]:  
\[\left[ \begin{array}{c}x_{UIT}\\ y_{UT}\end{array} \right]=-\left[ \begin{array}{c}(x_{G}-x_{P1})(y_{G}-y_{P1}) \\ (x_{G}-x_{P2})(y_{G}-y_{P2})\end{array} \right]^{-1}\cdot\left\{\left[ \begin{array} []{c}\Delta d_{1}\\ \Delta d_{2}\end{array} \right]\cdot d_{G}+\frac{1}{2}\cdot\left[ \begin{array} []{c}\Delta d_{1}^{2}+K_{P1}-K_{G}\\ \Delta d_{2}^{2}+K_{P2}-K_{G}\end{array} \right]\right\}.\] (5a) where \[ \begin{array}{c}K_{G}=x_{G}^{2}+y_{G}^{2},\\ K_{P1}=x_{P1}^{2}+y_{P1}^{2},\\ K_{P2}=x_{P2}^{2}+y_{P2}^{2}.\end{array}  \tag{5b}\]  
Equation (5a) can be written in a simpler form:  
\[\left[ \begin{array}{c}x_{UIT}\\ y_{UT}\end{array} \right]=-[W]\cdot d_{G}-[Z],\] (6a) in which under ( 5a ): \[[W]=\left[ \begin{array}{c}w_{1}\\ w_{2}\end{array} \right]=-\left[ \begin{array}{c}(x_{G}-x_{P1})(y_{G}-y_{P1}) \\ (x_{G}-x_{P2})(y_{G}-y_{P2})\end{array} \right]^{-1}\cdot\left[ \begin{array}{c }\Delta d_{1}\\ \Delta d_{2}\end{array} \right],\] (6b) and \[[Z]=\left[ \begin{array}{c}z_{1}\\ z_{2}\end{array} \right]=\frac{1}{2}\left[ \begin{array}{c}(x_{G}-x_{P1})(y_{G} -y_{P1})\\ (x_{G}-x_{P2})(y_{G}-y_{P2})\end{array} \right]^{-1}\cdot\left[ \begin{array}{ c}\Delta d_{1}^{2}+K_{P1}-K_{G}\\ \Delta d_{2}^{2}+K_{P2}-K_{G}\end{array} \right]. \tag{6c}\]  
As a result, a simple square equation of form must be solved to calculate the distance:  
\[ad_{\text{G}}^{2}+bd_{\text{G}}+c=0,\] (7a) whereby \[ \begin{array}{l}a=w_{1}^{2}+w_{2}^{2}-1,\\ b=-2\cdot(w_{1}\cdot x_{G}+w_{2}\cdot y_{G}+w_{1}\cdot z_{1}+w_{2}\cdot z_{2}),\\ c=x_{G}^{2}+y_{G}^{2}+2\cdot x_{G}\cdot z_{1}+2\cdot y_{G}\cdot z_{2}+z_{1}^{2 }+z_{2}^{2}.\end{array}  \tag{7b}\]As known, Equation (7a) has two solutions, the solution being defined by a relationship:  
\[d_{G1}=\frac{-b-\sqrt{b^{2}-4\cdot a\cdot c}}{2\cdot a}, \tag{7c}\]  
should be used when the localized terminal is located inside the area covered by reference station locations. The solution is determined by a relationship:  
\[d_{G1}=\frac{-b+\sqrt{b^{2}-4\cdot a\cdot c}}{2\cdot a}, \tag{7d}\]  
and should be applied outside the area defined above. The distance calculated in this way \(d_{G}\) should be inserted into relation (5a) and then used to calculate the desired coordinates \((x_{UT},y_{UT})\) of the localized user terminal (UT) [19]. The Foy algorithm is based on the linearization of the system of equations expressed by dependency (2), which--as already written above--should be developed in the Taylor series and then applied in an iterative approach. The result is an approximate solution, whose accuracy results from the number of iterative steps adopted. In the first step, rough initial values of the location of the localized terminal (UT) are defined by \((x_{UT},y_{UT})\).