## 2 Theoretical Background
### Proposed Methods for Estimating Side Gusts  
Assuming a coordinated flight without sideslip, the crosswind component along the aircraft's lateral axis can be estimated based on the ground speed component measured along the \(y_{B}\)-axis of the aircraft's body-fixed coordinate system (GS\({}_{y}\), Figure 2). A positive V\({}_{g}\) value indicates wind blowing from the left side of the aircraft (opposite to the direction of the \(y_{B}\)-axis).  
\[V_{g-lo}\ =GS_{y}\ =GS\cdot\sin(\mathrm{TRK}-\mathrm{HDG}_{\mathrm{TRUE}}) \tag{33}\]  
In the case of analyzing the lateral gust component, the sideslip angle can play a significant role. Figure 5 illustrates the relationships between the TAS vector measured in the aircraft-fixed coordinate system (F\({}_{\mathrm{B}}\)), the AIRSPEED vector representing the actual velocity of the aircraft in the air-mass-fixed coordinate system (refe  
Figure 4: Bode diagram (amplitude and phase) for linear CF.  
and the GS velocity relative to the ground (reference frame \(\rm F_{v}\)). By considering the angle of sideslip (AOS), the sideslip airspeed (SAS) can be determined:  
\[SAS=TAS\cdot\sin(AOS) \tag{34}\]  
By incorporating relationships (33) and (34), the value of the gust \(\rm V_{g}\) can be calculated using the following equation:  
\[V_{g-lo}\ =\rm GS_{y}-SAS=GS\cdot\sin(TRK-HDG_{TRUE})-TAS\cdot\sin(AOS) \tag{35}\]  
Determining the \(\rm GS_{y}\) component requires measuring GS, track angle (TRK), and the aircraft's heading (PSI), as in the case of longitudinal gust estimation. To calculate the lateral velocity in a sideslip, it is necessary to measure the angle of sideslip (AOS) and true airspeed (TAS). Similar to the previously discussed estimation methods, to enhance the accuracy of estimations during sudden and turbulent gusts, the use of a high-frequency signal was also adopted. For this purpose, the lateral acceleration component provided by accelerometers integrated into the ADIRU systems, which are part of the aircraft's onboard systems, was utilized. While this signal is not suitable for estimating the total value of the crosswind, integrating the lateral acceleration enables relatively precise determination of the component for sudden and short-lived lateral gusts.  
In the continuation of work on developing a three-dimensional gust estimation algorithm, including lateral gusts, it was decided to retain the use of complementary filtering, which yielded the most favorable results during the studies on headwind gust estima  
Figure 5: Illustration of relations from Equation (27).  
tion [28]. Based on dependencies (25) and (35), the complete law of sidewind estimation (including temporary gusts) can be formulated as Equations (36) and (37).  
\[V_{g-est\_CF}=\frac{1}{T_{cv}\cdot s+1}\cdot V_{g-lo}+\frac{T_{cv}\cdot s}{T_{cv }\cdot s+1}\cdot v_{g-hi} \tag{36}\]  
\[\begin{array}{l}V_{g-est\_CF}=\frac{1}{T_{cv}\cdot s+1}\cdot(GS\cdot\sin( \text{TRK}-\text{HDG}_{\text{TRUE}})-TAS\cdot\sin(AOS))+\\ +\frac{T_{cv}\cdot s}{T_{cv}\cdot s+1}\cdot(v+k_{1v}\cdot v+k_{2v}\cdot p+k_{3v }\cdot r+k_{4v}\cdot\delta_{A}+k_{5v}\cdot\delta_{R}+k_{6v})\end{array} \tag{37}\]  
After assuming \(v=\frac{1}{s}\cdot a_{y}\), neglecting constant values on the input of hi-pass filter, as well as the model parameters that depend on the flight state, this can be expressed by approximation (38).  
\[\begin{array}{l}V_{g-est\_CF}\approx\frac{1}{T_{cv}\cdot s+1}\cdot(GS\cdot \sin(\text{TRK}-\text{HDG}_{\text{TRUE}})-TAS\cdot\sin(AOS))+\\ +\frac{T_{cv}\cdot s}{T_{cv}\cdot s+1}\cdot(v+k_{1v}\cdot v+k_{2v}\cdot p+k_{3v }\cdot r+k_{4v}\cdot\delta_{A}+k_{5v}\cdot\delta_{R}+k_{6v})\end{array} \tag{38}\]