Axis Simulator — Wind / Body / Earth

BODY ATTITUDE (NED)

ψ Yaw 0.0°

θ Pitch 0.0°

φ Roll 0.0°

BODY→WIND

α AOA 0.0°

β AOS 0.0°

PRESET:

ψ 0°

θ 0°

φ 0°

EARTH↔BODY

AOA→EULER

INFO

EULER ANGLES (3-2-1)

ψ Yaw / Heading

θ Pitch

φ Roll

PRESETS (ψ / θ / φ)

BODY Xb DIRECTION (NED FRAME)

N (North)1.000

E (East)0.000

D (Down)0.000

INPUT: AERODYNAMIC ANGLES

α AOA — Angle of Attack

β AOS — Sideslip Angle

SOLUTION 1 — Φ = 0 (No Bank)

◈ Φ=0 SOLUTION: (Ψ, Θ)

φ Roll 0.000 °

ψ Yaw —

θ Pitch —

SOLUTION 2 — Ψ = 0 (Level Flight)

◈ Ψ=0 SOLUTION: (Φ, Θ)

ψ Yaw 0.000 °

φ Roll —

θ Pitch —

Inverse solution formulas:

Solution 1 (φ=0):
ψ = −β
θ = α

Solution 2 (ψ=0):
φ = atan2(sinβ, sinα·cosβ)
θ = arccos(cosα·cosβ)

Verification: C_EB·[1,0,0]ᵀ
→ u=cosαcosβ, v=sinβ, w=sinαcosβ

AXIS LEGEND

EARTH (NED) FRAME

Xe — North

Ye — East

Ze — Down

BODY FRAME

Xb — Forward (Nose)

Yb — Right (Starboard)

Zb — Down

WIND FRAME

Xw — Drag (V̂)

Zw — Lift (−Zw)

Yw — Side Force

Wind is constant, body rotates.

3-2-1 Euler (Tait-Bryan):
C_EB = Rx(φ)·Ry(θ)·Rz(ψ)

Velocity in body frame:
[u,v,w] = C_EB · V_wind,NED

Exact AOA/Sideslip:
α = atan2(w, u)
β = asin(v / |V|)

Wind NED vector:
V_w = [cosα₀cosβ₀, sinβ₀, sinα₀cosβ₀]