4.2.1. Definition of a trihedron or frame of reference

The association of the space frame and the time frame constitutes a frame of reference (R).

In the remainder of the book, we will note, for simplicity, the frame of reference (R) as follows:

4.2.2. Galilean trihedron

In physics, a Galilean or inertial frame of reference is a frame in which an isolated object (on which no force is exerted or on which the resultant of the forces is zero) is in uniform rectilinear translational movement (immobility being a particular case of uniform rectilinear movement): the speed of the body is constant (over time) in direction and in norm.

This means that the principle of inertia, which is stated in Newton’s first law – “every body perseveres in the state of rest or of uniform straight-line motion in which it finds itself, unless some force acts upon it and compels it to change its state” – is true.

A Galilean referential is thus called as such as a tribute to Galileo and more particularly to Galilean relativity.

The search for an inertial frame of reference is a delicate subject; the concrete determination of such a frame of reference is always approximate.

In a non-inertial frame of reference, which is animated by an accelerated movement relative to a Galilean frame of reference, inertial forces must intervene.

4.2.3. Absolute trihedron

We assume that there are at least three stars, E1, E2 and E3, considered as points, which, together with the center of mass S of the solar system, form an undeformable tetrahedron.

Let be an orthonormal coordinate system rigidly linked to this tetrahedron. By providing this frame of reference with a time base, we constitute the Kepler frame of reference (K), which we assume to be Galilean.

This trihedron is also called the Copernicus or Kepler trihedron.

4.2.4. Local geographic trihedron

The Earth being assimilated to a sphere with center G, the frame of reference (LGT) is constituted using the frame of reference (OX_T,OY_T,OZ_T) defined as follows:

• – O is a point on the surface of the Earth;
• – OX_T is tangent to the meridian and directed towards the north;
• – OY_T is tangent to the parallel of O and directed towards the east;
• – OZ_T is directed towards the center of the Earth.

4.2.5. Terrestrial trihedron

The Earth being assimilated to a sphere of center G, the terrestrial reference frame (TER) is defined as follows:

• – G is the center of the Earth;
• – G_xT is tangent to the meridian and directed towards the north;
• – G_yT is tangent to the parallel of G and directed towards the east;
• – G_zT completes this direct trihedron (directed towards the center of the Earth).

4.2.6. Aircraft trihedron

The aircraft trihedron (drone, helicopter, plane, etc.), or Lilienthal axis system (G,X_e,Y_e,Z_e), is defined as follows:

• – the trihedron has an origin G, which is the center of gravity of the aircraft;
• – GX_e is the longitudinal axis, which is carried by the axis of the aircraft towards the nose of the aircraft;
• – GY_e is the axis perpendicular to the plane of symmetry, oriented to the right (pilot direction);
• – GZ_e ┴ GX_e is located in the plane of symmetry of the aircraft and oriented ˃ 0 towards the belly of the aircraft (towards the center of the Earth); it completes the direct trihedron.

4.2.7. Aircraft aerodynamic speed trihedron (G, X_vae, Y_vae, Z_vae)

The aerodynamic speed trihedron of the aircraft (G, X_vae, Y_vae, Z_vae) is defined as follows:

• – the trihedron has an origin G, which is the center of gravity of the aircraft;
• – GX_vae is carried by the aerodynamic speed;
• – GZ_vae ┴ GX_vae is located in the plane of symmetry of the aircraft and oriented ˃ 0 towards the belly of the aircraft (towards the center of the Earth);
• – GY_vae completes the direct trihedron.

4.2.8. Balance trihedron

The coefficients on tracks y (GY_b)and z (GZ_b) are restored in an aircraft trihedron demodulated of the aerodynamic roll. The balance trihedron is defined as follows (G,X_b,Y_b,Z_b):

• – the trihedron has an origin G, which is the center of measurement of the balance;
• – GX_b is carried by the X axis of the aircraft;
• – GZ_b is located in the plane of symmetry of the aircraft and oriented ˃ 0 towards the belly of the aircraft (towards the center of the Earth);
• – GY_b completes the direct trihedron.

4.4.1. Aircraft trihedron (G, X₁, Y₁, Z₁) with respect to the reference trihedron (G, X₀, Y₀, Z₀)

This position is defined by the coordinates of the center of gravity G and the relative orientation of the axes. From the coordinates of G, only the altitude Z is kept. The relative position is defined by Figure 4.3. We used, to define the angles, the trihedron (G,X_h,Y_h,Z₀), where GX_h is the projection of GX₁on the horizontal plane passing through G:

• – Ψ: the angle defining the azimuth (or the heading if OX₀ is directed towards the north);
• – θ: the longitudinal attitude angle (also called the attitude);
• – ϕ: the roll angle.

We still use the angles θ₁ and ϕ₂. (G,X,Z) is the intersection of the plane of symmetry(G,X₁,Z₁) with the horizontal plane of G:

• – θ₁: the pitch angle;
• – ϕ₂: the angle of lateral attitude (angle of GY₁with the horizontal plane not shown).

4.4.2. Aerodynamic trihedron of the aircraft (G, X_va, Y_va, Z_va) with respect to the aircraft trihedron (G, X_e, Y_e, Z_e)

The aerodynamic slope is defined by the sideslip angle β and angle of attack α:

• – β: the angle of the speed vector with the plane of symmetry;
• – α: the angle of the projection of the speed vector on the plane of symmetry with the longitudinal reference of the aircraft.

We define the slope as the angle of the velocity with the horizontal plane, counted positively when GX is located above the horizontal plane (see Figures 4.1 and 4.2; in Figure 4.2, the angle α is denoted as i and the angle β is denoted as j).

4.5.1. Position of the aircraft trihedron with respect to the local geographical trihedron

The reference trihedron is the local geographic trihedron defined at the point of projection of the aircraft on the launch ramp (G,X_lgt,Y_lgt,Z_lgt).

For the change of trihedron, we carry out the three classic rotations:

• – of the angle Ψ around the axis Z_lgt, which brings GX_lgt into GX and GY_lgt into GY₁, in the vertical plane passing by GZ_lgt;
• – of the angle θ around the axis Y1, which brings GY₁ into GX_eand GZ_lgt into GZ₁;
• – of the angle ϕ around the axis GX_e, brings GY₁ into GY_e and GZ₁ into GZ_e.

Consequently, the coordinates (X_lgt, Y_lgt, Z_lgt) in the local geographic trihedron are written using the following transfer matrix from the coordinates (X_e, Y_e, Z_e) in the aircraft trihedron:

and:

with:

The inverse matrix, which makes it possible to pass from the aircraft trihedron to the local geographical trihedron, is symmetrical compared to the diagonal:

where:

M₁₁ = cos Ψ ∗ cos θ

M₁₂ = − sin Ψ ∗ cos ϕ + cos Ψ ∗ sin θ ∗ sin ϕ

M_1ଷ = sin Ψ ∗ sin ϕ + cos Ψ ∗sin θ ∗ cos ϕ

M₂₁ = sin Ψ ∗ cos θ

M₂₂ = cos Ψ ∗ cos ϕ + sin Ψ ∗ sin θ ∗ sin ϕ

M₂₃ = − cos Ψ ∗ sin ϕ + sin Ψ ∗ sin θ ∗ cos ϕ

M₃₁ = − sin θ

M₃₂ = cos θ ∗ sin ϕ

M₃₃ = cos θ ∗ cos ϕ

4.5.2. Position of the aerodynamic trihedron with respect to the terrestrial trihedron

The reference trihedron is the earth trihedron redefined with respect to the center of the Earth (G, X₀, Y₀, Z₀).

The aerodynamic trihedron (G, X_A, Y_A, Z_A) is usually identified by three classical angles compared to the reference trihedron.

For the change of trihedron, we carry out the three classic rotations:

• – the aerodynamic route or the aerodynamic azimuth χ_a: rotation around axis Z₀, which brings GX₀ into GX_h and GY₀ into GY_h, in the vertical plane passing from GZ₀;
• – the aerodynamic slope γ_a around the axis GY_h, which brings GX_h into GX_a and GZ₀ into GZ_f;
• – the rotation μ_a around the axis GX_a, which brings GY_h into GY_a and GZ_f into GZ_a.

Therefore, the coordinates (X_A, Y_A, Z_A) in the terrestrial trihedron are written using the following transfer matrix from the coordinates (X₀, Y₀, Z₀) in the terrestrial trihedron:

and:

with:

The inverse matrix, which makes it possible to pass from the aircraft trihedron to the terrestrial trihedron, is symmetrical with respect to the diagonal:

where:

M₁₁ = cos χ_a ∗ cos γ_a

M₁₂ = − sin χ_a ∗ cos μ_a + cos χ_a ∗ sin γ_a ∗ sin μ_a

M₁₃ = sin χ_a ∗ sin μ_a + cos χ_a ∗ sin γ_a ∗ cos μ_a

M₂₁ = sin χ_a ∗ cos γ_a

M₂₂ = cos χ_a ∗ cos μ_a + sin χ_a ∗ sin γ_a ∗ sin μ_a

M₂₃ = − cos χ_a ∗ sin μ_a + sin χ_a ∗ sin γ_a ∗ cos μ_a

M₃₁ = − sin γ_a

M₃₂ = cos γ_a ∗ sin μ_a

M₃₃ = cos γ_a ∗ cos μ_a

4.5.3. Position of the aircraft trihedron in relation to the aerodynamic speed trihedron

The aerodynamic speed trihedron of the aircraft (G, X_va, Y_va, Z_va) is defined as follows:

• – GX_va is driven by the aerodynamic speed;
• – GZ_va ┴ GX_va is located in the plane of symmetry of the aircraft and oriented ˃ 0 towards the belly of the aircraft (towards the center of the Earth);
• – GY_va completes the direct trihedron.

Axis GZ_va being located by definition in the aircraft symmetry plane (G, X_e, Z_e), just two angles are enough to achieve the change of reference:

• – the incidence α is the angle of the axis (GX_e) with the plane (G, X_va, Y_va) of the aerodynamic trihedron;
• – the slip β is the angle of the axis (GX_va) with the plane of symmetry (G, X_e, Z_e).

Moreover:

• – α ˃ 0 when (G,X_e) is above the plane;
• – β ˃ 0 if the projection of the speed vector is positive on the axis (G, Y_e).

Therefore, the coordinates (X₀, Y₀, Z₀) in the reference trihedron are written using the following transfer matrix from the coordinates (X_e, Y_e, Z_e) in the aircraft trihedron:

and:

with the matrix which makes it possible to pass from the aerodynamic speed trihedron to the aircraft trihedron:

The inverse matrix, which makes it possible to pass from the aircraft trihedron to the aerodynamic speed trihedron, is symmetrical with respect to the diagonal:

The total angle of attack is the angle αT between axis GX_e of the aircraft and axis GX_a of the aerodynamic speed.

4.5.4. Position of the aircraft trihedron in relation to the balance trihedron

The position of the aircraft trihedron (G, X_e, Y_e, Z_e) with respect to the balance trihedron (G, X_b, Y_b, Z_b) is given by the equations in Figure 4.7.

Consequently, the coordinates (X_e,Y_e,Z_e)in the aircraft trihedron can be written thanks to the following transfer matrix from the coordinates (X_b, Y_b, Z_b) in the balance trihedron:

COMMENTS ON FIGURE 4.7.–

• – X_e = x_b;
• – Y_e = Y_b ∗ cos ϕ_a – Z_b ∗ sin ϕ_a;
• – Z_e = Y_b ∗ (− sin ϕ_a) + Z_b ∗ cos ϕ_a.

The inverse matrix (matrix)^–1, which makes it possible to pass from the balance trihedron to the aircraft trihedron, is symmetrical with respect to the diagonal:

The flight coefficients are given by the following tables:

CX_b = Cx_e

CY_b = Cy_e ∗ cos ϕ_e – CZ_e ∗ sin ϕ_e

CZ_b = Cy_e ∗ sin ϕ_e + Cz_e ∗ cos ϕ_e

CL_b = CL_e

CM_b = CM_e ∗ cos ϕ_e – CN_e ∗ sin ϕ_e

CN_b = CM_e ∗ sin ϕ_e + CN_e ∗ cos ϕ_e

Cx_e = Cx_b

Cy_e = Cy_b ∗ cos ϕ_e + Cz_b ∗ sin ϕ_e

Cz_e = Cy_b ∗ (−sin ϕ_e) + Cz_b ∗ cos ϕ_e

CL_e = CL_b

CM_e = CM_b ∗ cos ϕ_e + CN_b ∗ sin ϕ_e

CN_e = CM_b ∗ (−sin ϕ_e) + CN_b ∗ cos ϕ_e

The aerodynamic coefficients obtained in the wind tunnel are the coefficients generally defined in [0°, 45°].

4.5.5. Position of the terrestrial trihedron in relation to the local geographic trihedron

The meanings of the parameters used are as follows:

• – Ω: the speed of rotation of the Earth with respect to an absolute reference;
• – λ_M: the difference between the latitude of the aircraft and the latitude of the origin of the test track;
• – λ_ct: the latitude of the origin of the test track;
• – G_M: the difference between the longitude of the aircraft and the longitude of the origin of the test track.

The position of the aircraft in the land reference is given by the following transfer matrix using two successive rotations. The axis (GZ_lgt) being located by definition in the aircraft plane of symmetry (G, X_e, Z_e), just two angles are enough to achieve the change of reference:

• – the latitude (λ_M + λ_ct) is the angle of the axis (GX_e) reduced to (GX_lgt) with the plane (G, X_ter, Y_ter) of the aerodynamic trihedron;
• – the longitude (G_M) is the angle of the axis (GX_e) reduced to (GX_lgt) with the plane of symmetry (G, X_ter, Y_ter).

When (GX_lgt)is above the plane, (λ_M + λ_ct) ˃ 0. If the projection of the speed vector is positive on the axis (GY_lgt), (G_M) ˃ 0.

Therefore, the coordinates (X_ter, Y_ter, Z_ter) d in the terrestrial trihedron are written using the following transfer matrix from the coordinates (X_lgt, Y_lgt, Z_lgt) in the local geographic trihedron:

The matrix that makes it possible to pass from the terrestrial trihedron to the local geographical trihedron is therefore:

The inverse matrix that makes it possible to pass from the local geographical trihedron to the terrestrial trihedron is symmetrical relative to the diagonal: