Chapter 2 Coordinate frames

Inertial frame and vehicle frame

1. Vehicle frame has same orientation as inertial frame.
2. Vehicle frame is fixed at CG of aircraft
3. Inertial and vehicle frames are referred as NED frames

Vehicle-1 frame

1. Rotation around z-axis of vehicle frame
2. This rotation is defined as the heading angle

Rotation from vehicle to vehicle-1 frame is defined as:

$$R_v^{v_1}(\psi )=\left[\begin{array}{ccc}\cos \psi & \sin \psi & 0\\ -\sin \psi & \cos \psi & 0\\ 0& 0& 1\end{array}\right]$$

Vehicle-2 frame

1. Rotation around y-axis of vehicle-1 frame
2. This rotation is defined as the pitch angle
3. Euler angle singularity at $90^o$ also called gimbal lock

Rotation from vehicle-1 frame to vehicle-2 frame is defined as:

$$R_{v_1}^{v_2}(\psi )=\left[\begin{array}{ccc}\cos \theta & 0& -\sin \theta \\ 0& 1& 0\\ \sin \theta & 0& \cos \theta \end{array}\right]$$

Body frame

1. Rotation around x-axis of vehicle-2
2. This rotation is called roll angle

Rotation from vehicle-2 frame to body frame is defined as:

$$R_{v_2}^b(\varphi )=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \varphi & \sin \varphi \\ 0& -\sin \varphi & \cos \varphi \end{array}\right]$$

Stability frame

This frame helps us rigorously define angle of attack and is useful for analyzing stability of aircraft.

Rotation from body frame to stability frame is defined as:

$$R_b^s(\alpha )=\left[\begin{array}{ccc}\cos \alpha & 0& \sin \alpha \\ 0& 1& 0\\ -\sin \alpha & 0& \cos \alpha \end{array}\right]$$

Wind frame

1. Wind frame is equal to flow frame in Fossen
2. Wind frame helps us rigorously define side-slip angle
3. Side-slip angle is nominally zero for tailed aircraft

Rotation from stability frame to wind frame is define as:

$$R_s^w(\beta )=\left[\begin{array}{ccc}\cos \beta & \sin \beta & 0\\ -\sin \beta & \cos \beta & 0\\ 0& 0& 1\end{array}\right]$$

Airspeed, wind speed and ground speed

$$V_a=\text{airspeed}$$ $$V_w=\text{windspeed}$$ $$V_g=\text{groundspeed}$$

$$V_a=V_g-V_w$$ $$V_g^b=\left[\begin{array}{c}u\\ v\\ w\end{array}\right]\to \text{speed over ground in body frame}$$ $$V_w^b=\left[\begin{array}{c}{u}_w\\ v_w\\ w_w\end{array}\right]=R_v^b(\varphi ,\theta ,\psi )\left[\begin{array}{c}{w}_n\\ w_e\\ w_d\end{array}\right]\to \text{windspeed in body}$$ $$V_a^w=\left[\begin{array}{c}{V}_a\\ 0\\ 0\end{array}\right]\to \text{wind frame air speed}$$

Airspeed, angle of attack and sideslip angle

$$V_a^b=\left[\begin{array}{c}{u}_r\\ v_r\\ w_r\end{array}\right]=R_w^b\left[\begin{array}{c}{V}_a\\ 0\\ 0\end{array}\right]\to \text{relative velocity in body (airspeed)}$$ $$\left[\begin{array}{c}{u}_r\\ v_r\\ w_r\end{array}\right]=V_a\left[\begin{array}{c}\cos \alpha \cos \beta \\ \sin \beta \\ \sin \alpha \cos \beta \end{array}\right]\to \text{airspeed in body}$$ $$V_a=\sqrt{u_r^2+v_r^2+w_r^2}\to \text{amplitude in airspeed}$$ $$\alpha ={\tan }^{-1}\left(\frac{w_r}{u_r}\right)\to \text{angle of attack}$$ $$\beta ={\sin }^{-1}\left(\frac{v_r}{\sqrt{u_r^2+v_r^2+w_r^2}}\right)\to \text{sideslip}$$

Course and flight path angles

Wind triangle horizontal projection

Wind triangle vertical plane