*3.1. Reference Frames*

Three right-handed orthogonal reference frames are introduced, according to the definitions in [25]:


4. an aircraft reference frame, <sup>F</sup>*<sup>A</sup>* <sup>=</sup> {*OA*; *<sup>x</sup>A*, *<sup>y</sup>A*, *<sup>z</sup>A*}, used to locate *CG* and all helicopter components: axes are parallel to the body-fixed frame axes, such that *x<sup>A</sup>* = −*xB*, *y<sup>A</sup>* = *yB*, and *z<sup>A</sup>* = −*zB*. The origin is located ahead and below the rotorcraft at some arbitrary point within the plane of symmetry. Stations (*ST*) are measured positive aft along the longitudinal axis. Buttlines (*BL*) are lateral distances, positive to the pilot's right, and waterlines (*WL*) are measured vertically, positive upwards. A sketch of the rotorcraft, including the selected F*<sup>A</sup>* frame, is reported in Figure 4. The positions of the main components, expressed in F*A*, are listed in Tables 1 and 2, together with relevant helicopter data.

**Figure 4.** Sketch of Zefhir helicopter (courtesy of Curti Aerospace Division).


**Table 1.** MR and TR relevant parameters.

**Table 2.** Fuselage, empennages, and miscellaneous components location.



Let *<sup>s</sup>*(·) = sin(·), *<sup>c</sup>* (·) = cos(·). Vector transformation between <sup>F</sup>*<sup>H</sup>* and <sup>F</sup>*<sup>B</sup>* is provided by the rotation matrix [12]

$$\mathbf{R}(\mathfrak{a}) = \begin{bmatrix} c\theta \ c\psi & c\theta \ s\psi & -s\theta \\ s\phi \ s\theta \ c\psi - c\phi \ s\psi & s\phi \ s\theta \ s\psi + c\phi \ c\psi & s\phi \ c\theta \\ c\phi \ s\theta \ c\psi + s\phi \ s\psi & c\phi \ s\theta \ s\psi - s\phi \ c\psi & c\phi \ c\theta \end{bmatrix} \tag{1}$$

obtained by a 3-2-1 Euler rotation sequence where *α* = [*φ*, *θ*, *ψ*] *<sup>T</sup>* describes the attitude of the rotorcraft in terms of the classical 'roll', 'pitch', and 'yaw' angles, respectively. The following notation is adopted: if *w* is an arbitrary vector, its components are transformed from <sup>F</sup>*<sup>H</sup>* to <sup>F</sup>*<sup>B</sup>* through *<sup>w</sup><sup>B</sup>* <sup>=</sup> *R wH*. In what follows, the subscript *<sup>B</sup>* will be dropped for simplicity.
