Lecture Notes: Reference Frames and Euler Angles

Inertial and Non-Inertial Frames

• Four primary reference frames used for EoM description: inertial and non-inertial frames.

  • Inertial frames: Frames where Newton's laws of motion apply directly without needing fictitious forces.

  • Earth Centered Earth Fixed (ECEF), also described as North-East-Down (NED).

  • Non-inertial frames: Frames that accelerate relative to an inertial frame, requiring fictitious forces (e.g., Coriolis) for Newton's laws to hold.

  • Generalised body-fixed frame (origin at CoG).

  • Stability frame.

  • Wind (aerodynamic) frame.

  • In practice (short-term flight): Earth is treated as inertial; aircraft motion described as perturbations about straight-and-level flight.

Euler Angles and Attitude Representation

• Attitude describes orientation of body axes relative to an inertial frame.

• Euler angles (φ, θ, ψ): Used to represent attitude.

  • φ = Roll (rotation about the x-axis)

  • θ = Pitch (rotation about the y-axis)

  • ψ = Yaw (rotation about the z-axis)

  • Common 3-2-1 sequence: rotate about z (ψ), then y (θ), then x (φ). This specific order is standard in aerospace engineering, providing a unique and intuitive representation for aircraft attitude.

• Attitude transformation is captured by a Directional Cosine Matrix (DCM) or rotation matrices; order of multiplication matters.

  • DCMs are orthogonal matrices, meaning their inverse is equal to their transpose ($C^{-1} = C^T$). This property simplifies frame transformations.

• Transformations:

  • Velocity/inertial to body: $v<em>b = C</em>{bI} v<em>I$ (or $v</em>I = C<em>{Ib} v</em>b$)

  • If $C<em>{Ib}$ is the inverse of $C</em>{bI}$, and the matrix is orthogonal, then $C<em>{Ib} = C</em>{bI}^T$.

Frame Transformations and Directional Cosine Matrices

• Body frame to stability frame:

  • First rotation by -α about the body y-axis (to align with the stability frame).

  • $C<em>{bst} = R</em>y(-\alpha)$, where $R_y(\theta)$ is the elementary rotation matrix about the y-axis by angle $\theta$.

• Stability frame to wind frame:

  • Second rotation by β about the z-axis of the stability frame.

  • $C<em>{stw} = R</em>z(\beta)$, where $R_z(\phi)$ is the elementary rotation matrix about the z-axis by angle $\phi$.

• Body to wind (combined):

  • $C<em>{bw} = R</em>z(\beta) R_y(-\alpha)$.

• Relative frames and their connections are summarized with a sequence of Directional Cosine Matrices to relate body, stability, wind, and inertial frames.

Aerodynamic Frames: Stability and Wind

• Stability frame (st) is rotated from the body frame by the angle of attack α about the body y-axis.

• Wind frame is obtained by rotating the stability frame by the sideslip angle β about the z-axis of the stability frame.

• Aerodynamic forces (lift, drag) are resolved along the body axes using these frames.

• Key axes:

  • x-axis aligned with body forward direction

  • z-axis points down in NED convention (downward under gravity in inertial sense)

Velocity, Forces, and Moments Notation

• Body-frame velocity components:

  • U, V, W (along x, y, z of body frame).

• Angular velocity components in body frame:

  • p, q, r (roll, pitch, yaw rates in body axes).

• Aerodynamic forces in body frame:

  • $F<em>b$ with components (X, Y, Z) or ($F</em>x$, $F<em>y$, $F</em>z$).

• Aerodynamic moments in body frame:

  • M, N, L (moments about x, y, z axes) and corresponding angular rates p, q, r.

• Control surface rotations (sign conventions):

  • Elevator: positive down (in body frame).

  • Rudder: positive to the left.

  • Ailerons: positive right-down, left-up.

Angle of Attack and Sideslip

• True airspeed (in wind frame): $V_{\text{true}}$.

• Velocity components in body frame: (U, V, W).

• Angle of attack α: angle between the body's x-axis and the projection of the velocity onto the x-z plane.

  • Convention: $\alpha = \text{atan2}(W, U)$. The atan2 function correctly handles all four quadrants, providing the principal value of the argument of the complex number $U + iW$.

• Sideslip β: angle between the projection of velocity onto the x-y plane and the x-axis.

  • Convention: $\beta = \text{atan2}(V, U)$. Similarly, atan2 ensures correct quadrant resolution for the angle based on $U$ and $V$.

• For small angles (i.e., when $|\alpha|$ and $|\beta|$ are small, typically less than about 10-15 degrees), the approximations simplify to:

  • $\alpha \approx W/U$

  • $\beta \approx V/U$

Summary of Key Concepts

• Motions described in multiple frames: inertial (NED/ECEF/NEU), body-fixed, stability, and wind.

• Attitude described by Euler angles with 3-2-1 sequence; properly transform vectors between frames via DCMS.

• Stability and wind frames used to resolve aerodynamic forces into body-axis components.

• Velocity and attitude are linked through translational and rotational EoM; sign conventions for control surfaces dictate moments and forces.

• α and β quantify orientation of the flight path relative to the body axes; expressed in terms of body-frame velocity components.

Quick References

• Euler angles: φ (roll) about x, θ (pitch) about y, ψ (yaw) about z; 3-2-1 sequence.

• Attitude transforms: $v<em>I = C</em>{bI} v<em>b$, $v</em>b = C<em>{Ib} v</em>I$ with $C<em>{Ib} = C</em>{bI}^T$.

• Frame chain: body → stability (rotate by -α about y) → wind (rotate by β about z).