Relationship between earth frame attitude and acceleration for a quadcopter

Question

For a quadcopter, what is the relationship between roll, pitch, and yaw in the earth frame and acceleration in the x, y, and z dimensions in the earth frame? To be more concrete, suppose roll ($\theta$) is a rotation about the earth frame x-axis, pitch ($\phi$) is a rotation about the earth frame y-axis, and yaw ($\psi$) is a rotation about the z-axis. Furthermore, suppose $a$ gives the acceleration produced by all four rotors, i.e. acceleration normal to the plane of the quadcopter. Then what are $f, g, h$ in

$$a_x = f(a,\theta,\phi,\psi)$$ $$a_y = g(a,\theta,\phi,\psi)$$ $$a_z = h(a,\theta,\phi,\psi)$$

where $a_x$, $a_y$, and $a_z$ are accelerations in the $x$, $y$, and $z$ dimensions.

I've seen a number of papers/articles giving the relationship between x,y,z accelerations and attitude, but it's never clear to me whether these attitude angles are rotations in the earth frame or the body frame.

Answer 1

The orientation angles will be provided in the Earth frame (global or world frame is more appropriate). It is pretty much impossible to provide a frame orientation with angles with respect to that same frame.

Then, given some acceleration $a_b$ in the body frame, you can find the acceleration in the world frame $a_w$ by pre-multiplication with the rotation matrix $R$.

$a_w = R a_b$

If you assume a typical roll-pitch-yaw sequence, then this will look like:

$\begin{bmatrix} a_{w,x} \\ a_{w,y} \\ a_{w,z} \end{bmatrix} = \begin{bmatrix} c_\psi c_\theta & -s_\psi c_\phi + c_\psi s_\theta s_\phi & s_\psi s_\phi + c_\psi s_\theta c_\phi \\ s_\psi c_\theta & c_\psi c_\phi + s_\theta s_\psi s_\phi & -c_\psi s_\phi + s_\theta s_\psi c_\phi \\ -s_\theta & c_\theta s_\phi & c_\theta c_\phi \end{bmatrix} \begin{bmatrix} a_{b,x} \\ a_{b,y} \\ a_{b,z} \end{bmatrix}$

Where $c_x = \cos x$ and $s_x = \sin x$.

Of course, you can expand this to specify the x-y-z components of the world frame acceleration as individual equations as you mentioned:

$a_{w,x} = f(a_b, \phi, \theta, \psi)$

$a_{w,y} = g(a_b, \phi, \theta, \psi)$

$a_{w,z} = h(a_b, \phi, \theta, \psi)$

Note that I am using the convention where $\phi$ is roll (about x-axis), $\theta$ is pitch (about y-axis), and $\psi$ is yaw (about z-axis).

When the rotation sequence roll, pitch, yaw is applied, each rotation occurs with respect to a world frame axis. We could get the same final result by applying a yaw, pitch, roll sequence but about each new axis. This latter method is more intuitive when visualizing roll-pitch-yaw (or rather Euler angle) rotations.

Here is a figure to clarify the rotation sequence in case it helps. The world frame is shown in black, the body frame in red-green-blue, and intermediary frames during the "intuitive" rotation sequence are included in grey.