# Using Ground-Based Passive Reflectors for Improving UAV Landing
## 3 Algorithm Synthesis
### Principal Solution of the Navigation Problem in Onboard Radar  
The location finding method for the UAV during its landing is based on a range measurement location finding of an active user of the navigating system. Let us discuss in more details the two-step algorithm of the location finding for the UAV during its landing with the initial situation shown in Figure 6. A UAV, carrying an airborne radar, has a velocity vector \(\mathbf{V}_{H}\) and the true height \(H\), according to the autonomous navigation system, approaches the airport area. The reflectors placed on the ground are marked with numbers 1-4. Reflectors one and three are located on the rolling axis of the landing path and reflectors two and three are located on the orthogonal axis drawn through point \(O\), that is the intended point of the UAV touchdown. The first step of the procedure includes the emission of the probing signals; the receiving of its reflection from the previously positioned reflectors; the calculation of the distances to each reflector, denoted as \(R_{1}\), \(R_{2}\), \(R_{3}\), and \(R_{4}\). The vector of the distance measurements obtained as the output of the first step is the input data vector for the second step. The mathematical model of the input data vector can be described in the form of:  
\[\mathbf{y}_{R_{n},k}=R_{n}(x_{k},y_{k},z_{k})+\varepsilon_{R_{n},k} \tag{1}\]  
where \(R_{n,k}\) is the distance between the radar antenna and the \(n\)-th reflector at the observation moment; \(x_{k},y_{k}\), and \(z_{k}\) are the true current coordinates; \(\varepsilon_{R_{n},k}\) is the error in measuring the distance \(R_{n,k}\). In the Cartesian coordinate system, with the origin point located at the intended UAV touchdown point on the landing path, the distance between the airborne radar and the \(n\)-th reflector can be described as:  
\[y_{R_{n},k}=\sqrt{(X_{n}-x_{k})^{2}+(Y_{n}-y_{k})^{2}+(Z_{n}-z_{k})^{2}}. \tag{2}\]  
where \(X_{n}\), \(Y_{n}\) and \(Z_{n}\) are the coordinates of the \(n\)-th reflector. The vector of the current relative coordinates of the UAV at the \(k\)-th time step is estimated as the solution of the system of nonlinear equations; these equations are the equations of the spheres with reflectors in the centers and the radii equal to the distances to the airborne radar. The linearization procedure is invoked to simplify the system with a priory estimation taken from another system such as an autonomous UAV navigation system. After the appropriate linearization [51] is conducted, the presented system takes the form of:  
Figure 6: The principle of the estimation of the coordinated steps during the aircraft landing. \[\mathbf{\hat{x}}_{k}=\mathbf{\tilde{x}}_{k}+\left(\mathbf{\hat{H}}^{T}\mathbf{\hat{H} }\right)^{-1}\mathbf{\hat{H}}^{T}\mathbf{\Delta}_{y_{R,k}}, \tag{3}\]  
where \(\mathbf{\hat{x}}_{k}\) is the aircraft coordinate vector estimation at the \(k\)-th moment of time, \(\mathbf{\hat{x}}_{k}\) is the prior estimation at the \(k\)-th step, \(\mathbf{\hat{H}}\) is the cosine matrix, and \(\mathbf{\Delta}_{y_{R,k}}\) is the vector of the estimation errors. Figure 7 shows a UAV in the \(k\)-th moment of time having the coordinates \((x_{k},y_{k})\) in a rectangular coordinate system defined by the orthogonal axes \(X\) and \(Y\) on a plane formed by the UAV's coordinates and the \(n\)-th corner reflectors (CR) \(CR_{n}\). The \(X\)-axis was oriented along the runway and all the corner reflectors were located symmetrically. The cosine matrix \(\mathbf{\hat{H}}\) has the same number of rows as the number of the reflectors and the same number of columns as the number of the estimated coordinates:  
\[\mathbf{\hat{H}}(\mathbf{\hat{x}}_{k})= \begin{bmatrix}-\cos\alpha_{1,k}&-\cos \beta_{1,k}&-\cos\gamma_{1,k}\\ -\cos\alpha_{2,k}&-\cos\beta_{2,k}&-\cos\gamma_{2,k}\\ -\cos\alpha_{3,k}&-\cos\beta_{3,k}&-\cos\gamma_{3,k}\\ -\cos\alpha_{4,k}&-\cos\beta_{4,k}&-\cos\gamma_{4,k}\end{bmatrix} .