# Anti-Collision System for Small Civil Aircraft
## 4 Accuracy of Determining the Position of Flying Objects by Anti-Collision System
### Methodology for Evaluating the Accuracy of the Anti-Collision System  
The assessment of the accuracy of the anti-collision system needs to be carefully analyzed. The main sources of possible errors include the geometric configuration of the communication network users, the quality of the parameters transmitted in the FOs' position report, and the network synchronization. The task is to assess the effect of distance measurement errors \(\Delta\) d1-4 on the determination of the position of the anti-collision system. In the simulations, the distance measurement errors \(\Delta\)d1-4 with a random number generator with a normal distribution are mimicked. The generated random numbers are added to the actual coordinates FO1-4. The parameters of the normalized normal distribution are as follows: E (x) = m = 0 is the mean and the variance V (x) = \(\sigma^{2}\) = 1. As detected,pseudo-distances from four broadcast sources and 12 random number generators are available. To set the size of the error, the constant k, multiplied by each generated number is used, where:  
\[\text{M[k.X]}=\text{k. M. [X]} \tag{28}\]  
\[\text{D[c.X]}=\text{c}^{2}.\ \text{D[X]}, \tag{29}\]  
where: M--mean value, D--dispersion, k--non-random variable (constant), X--random variable. The created program writes errors to the row matrix for each coordinate and each user separately:  
\[\text{X}_{\text{LOi}}=[\text{$\Delta$x$}_{1}^{\text{i}},\ \text{$\Delta$x$}_{2}^{\text{i}}, \text{..., $\Delta$x$}_{n}^{\text{i}}], \tag{30}\]  
\[\text{Y}_{\text{LOi}}=[\text{$\Delta$y$}_{1}^{\text{i}},\ \text{$\Delta$y$}_{2}^{\text{i}}, \text{..., $\Delta$y$}_{n}^{\text{i}}], \tag{31}\]  
\[\text{Z}_{\text{LOi}}=[\text{$\Delta$z$}_{1}^{\text{i}},\ \text{$\Delta$z$}_{2}^{\text{i}}, \text{..., $\Delta$z$}_{n}^{\text{i}}], \tag{32}\]  
where: i = 1,2,3,4 (user index);  
n is the length of the flight in seconds;  
X\({}_{\text{LOi}}\)--row matrix of randomly generated errors of x coordinate of user FOi;  
Y\({}_{\text{LOi}}\)--row matrix of randomly generated errors of y coordinate of user FOi;  
Z\({}_{\text{LOi}}\)--row matrix of randomly generated errors of z coordinate of user FOi. Using the scalar sum operation of matrix elements (27) and matrix elements (30) to (32), we obtain new row matrices, the elements of which represent the X, Y, Z coordinates of each FO with error. \[\text{LOi X}=\text{FOi}\text{a+ X}\text{LOi}=[\text{a}_{1}^{\text{i}}+\text{$ \Delta$x$}_{1}^{\text{i}},\text{a}_{2}^{\text{i}}+\text{$\Delta$x$}_{2}^{ \text{i}},\text{...,$a$}_{n}^{\text{i}}+\text{$\Delta$x$}_{n}^{\text{i}}], \tag{33}\]  
\[\text{LOi Y}=\text{FOi}\text{b+ Y}\text{LOi}=[\text{b}_{1}^{\text{i}}+\text{$ \Delta$y$}_{1}^{\text{i}},\text{b}_{2}^{\text{i}}+\text{$\Delta$y$}_{2}^{ \text{i}},\text{...,$b$}_{n}^{\text{i}}+\text{$\Delta$y$}_{n}^{\text{i}}], \tag{34}\]  
\[\text{LOi Z}=\text{FOi}\text{c+ Z}\text{LOi}=[\text{c}_{1}^{\text{i}}+\text{$ \Delta$z$}_{1}^{\text{i}},\text{c}_{2}^{\text{i}}+\text{$\Delta$z$}_{2}^{ \text{i}},\text{...,$c$}_{n}^{\text{i}}+\text{$\Delta$z$}_{n}^{\text{i}}], \tag{35}\]  
where:  
LOi X-row matrix of coordinate X of user FOi with error \(\Delta\) xi;  
LOi Y-row matrix of coordinate Y of user FOi with error \(\Delta\) yi;  
LOi Z-row matrix of coordinate Z of user FOi with error \(\Delta\) zi. To calculate the pseudo-range to an unknown user LO5 with ACS, the geocentric coordinates from Equation (27) and the coordinates of other users FO1-4 from matrices (33) to (35) will be used.