# What is the relation between body-frame, body-fixed-frame and vehicle-frame?

When I first came across these three terms, I thought they were all the same. A frame attached to the body of a plane and centered at its center-of-mass with the frame's $$\left\{\hat{\mathbf{i}},\hat{\mathbf{j}},\hat{\mathbf{k}}\right\}$$ aligns with the plane's { nose, wings, belly } respectively. If the plane starts out heading north and yaws $$90^{\circ}$$, the new $$\hat{\mathbf{i}}$$ will then point East. Since the body-frame is fixed to the body of the plane, we can also called it body-fixed-frame. The body of Micro Air Vehicle (MAV) is termed as a vehicle instead of a plane, and thus the name vehicle-frame.

$$\text{body-frame} \equiv \textrm{body-fixed-frame} \equiv \textrm{vehicle frame}$$

I have always treated these terms as context-dependent synonyms until I see this following equation in my textbook (Small Unmanned Aircraft, Beard) where it rotates the body-frame to vehicle-frame! What? "Body" and "vehicle" are different? I couldn't make sense out of it. Since this equation is formulated on translational motion, I can treat the plane (or the vehicle) as a particle positioned at its center-of-mass (or origin of the body-frame). Any instantaneous translational velocity experienced by this particle interpreted by it's own body-frame is $$(\mathbf{v}^b_x,\mathbf{v}^b_y,\mathbf{v}^b_z)\equiv (u,v,w)$$. If we want to interpret these velocities under the inertial reference frame $$(\dot{p}_n,\dot{p}_e,\dot{p}_d)$$，shouldn't we just rotate the body-frame to the inertial-frame instead?

$$\frac{d}{dt} \; \left( \begin{array}{ c } p_n \\ p_e \\ p_d \end{array} \right) = \mathcal{R}^i_b \left( \begin{array}{ c } u \\ v \\ w \end{array} \right)$$

What does it mean to rotate a body-frame to a vehicle-frame ... isn't a body already a vehicle? And if rotating a body-frame to a vehicle-frame gives us values projected onto the inertial North-East-Down axis as the textbook suggest, wouldn't that makes the vehicle-frame aligned with the inertial-frame?

I've tried googling but with no success, and these terms are also not clearly defined in the textbook.

• Just to get this right, you are asking in reference to the transformation $\mathcal{R}^v_b$ depicted in your picture, right? Because the text of the textbook does not mention vehicle frame...I would say that this is a typo. It must be meant $\mathcal{R}^i_b$ because then the equation from your textbook is quite familiar... Dec 7, 2022 at 22:22
• @U_flow yep get it right - why the book goes with $\mathcal{R}^v_b$ instead of $\mathcal{R}^i_b$.
– KMC
Dec 8, 2022 at 0:39

$$\mathcal{F}^i$$ is the earth NED frame
$$\mathcal{F}^v$$ is a NED frame centered at the vehicle CG
$$\mathcal{F}^b$$ is a uvw frame centered at the vehicle CG
$$\mathcal{F}^{v1}$$ is the frame after the first Euler rotation ($$\psi$$)
$$\mathcal{F}^{v2}$$ is the frame after the second Euler rotation ($$\theta$$)
$$\mathcal{F}^{v3}$$ is the frame after the third Euler rotation ($$\phi$$), ie. $$\mathcal{F}^{v3} \equiv \mathcal{F}^b$$