# GNSS Performance Modelling and Augmentation for Urban Air Mobility +
## 3 GNSS Performance Characterization
### Error Modelling
#### 3.1.7 Airframe Reflection  
Signal reflections from the UAV frame is specific to each aircraft. An exhaustive approach is to apply a numerical technique such as the Method of Moments to a given airframe. Alternatively, in [15], a Physical Optics (PO) - based ray-tracer was applied to develop a model of fuselage reflection for large fixed-wing aircraft.  
The most obvious differences between the large-fixed wing aircraft scenario and present-day rotary wing aircraft lie in their physical dimensions, antenna placement configuration and material properties. Receiver antennas are typically sited 5-10 cm above the airframe, and are installed with a small ground plane to dampen the radiation pattern back lobe. This aids the minimization of the antenna gain for signals arriving from low elevation angles (such as signals reflected from the airframe). This is illustrated in Figure 8 for a microstrip patch antenna modelled with a dielectric substrate and small groundplane, as commonly installed on low-cost GNSS receivers. The figure shows the elevation gain pattern of the antenna for an azimuthal slice. It can be safely assumed that only the upper portion of the airframe contributes to the airframe multipath channel through a combination of reflection and scattering. Figure 9 depicts a reference platform that is broadly representative of most small to medium-sized UAV configurations on the market.  
The largest delay will be from a low elevation satellite (relative to the UAV body frame {B}) and can be derived through the same trigonometric relationships used to model ground reflection delay. Knowledge of the maximum dimension of a given UAV allows an approximation of the range of the parameters of the UAV.  
Figure 8: Microstrip patch antenna radiation pattern.  
Figure 9: Multipath region for a small UAV (owing to reflections from the aircraft body).  
possible echo delays from the body of the aircraft. A reference platform is adopted as shown in Figure 9 for the analysis.  
The height of the antenna above the mounting bed and the half-span (center to motor distance) is known (\(h\) and \(x\), respectively). The specular airframe reflection region (angular) is computed to be  
\[\theta\,\in\left[\tan^{-1}\frac{h}{x},\,90^{\circ}\right] \tag{21}\]  
where \(\theta\) is the elevation of the satellite in the UAV body frame {B}  
The corresponding reflection delays are bounded by:  
\[\delta\,\in\left[2h\sin\theta,2h\right] \tag{22}\]  
which simplifies to:  
\[\delta\,\in\left[\frac{h}{\sqrt{h^{2}+x^{2}}},2h\right] \tag{23}\]  
For example, assuming a span of 650 mm measured from propeller to propeller (most commercial small to medium-sized UAVs range in size from 450-650 mm), it can be seen from Figure 9 that specular reflections are restricted to satellites at elevations (in the UAV body frame) from 17\({}^{\circ}\) and above, leading to delays in the range of 19.49 ns - 66.66 ns. This corresponds to the short multipath region as designated by [26]. Nominal values of (\(\varepsilon_{\mathrm{r}}=3.5,\,\,\sigma=2\times 10^{-12}\)) corresponding to material parameters for polyamide thermoplastics commonly used in UAV airframes are assumed.