How to rotate a rotation quaternion in the body frame to a rotation quaternion in the world frame?

Question

I have a sensor (3 axis gyroscope) which can rotate and measure angular velocity in 3 dimensions (aligned with the sensor). I know what its current orientation is with respect to the world frame. Call this quaternion qs. I take a reading from my gyroscope and integrate it to give me a rotation in the sensor frame. Call this quaternion qr.

I now want to apply the rotation qr to the current orientation qs to obtain the new orientation, qs'. But I cannot use qr directly as it describes a rotation in the sensor body frame.

I need to transform my rotation quaternion into the world frame, and then I could just apply it to the orientation i.e. qs' = qs * qr_world. But I am really struggling to understand how I can perform this transformation qr -> qr_world.

Does this even make sense? I wonder if I have fundamentally misunderstood some concepts here. If it does make sense, then I am specifically interested in understanding how to do this using quaternion operations (if that is possible) rather than rotation matrices or euler angles.

Answer 1

As pointed out in my earlier comment, this is actually simpler than you may think. Remember, $qs$ and $qr$ are fundamentally different, where the former represents orientation (in reference to the outside world) and the latter represents rotation (irrespective of any reference coordinate system).

You're right in saying that you don't need to do any additional transformations and that your new orientation $qs'$ will be given by:

$qs' = qs \times qr$

where;

• $qs$ is the previous position, and
• $qr$ is a single rotation by a given angle $\theta$ about a fixed axis (obtained from the gyroscope).

Example

Let's presume a simple analogy with straight line displacement;

$\delta s = v \times \delta t$

Given an initial position $s$, it can be said that moving at a velocity $v$ for a period of time $t$ sees us ending up at a final position $s'$:

$s' = s + \delta s$

Similarly, presume angular displacement (rotation in one axis);

$\delta \theta = \omega \times \delta t$

Given an initial angular position $\theta$, it can be said that moving at an angular velocity $\omega$ for a period of time $t$ sees us ending up at a final angular position $\theta'$:

$\theta' = \theta + \delta \theta$

Now, presume angular displacement in three-dimensional space (rotation in three axes) that, according to Euler's rotation theorem, is equivalent to a single rotation by a given angle $\theta$ about a fixed axis $\hat \omega$;

$qr_w = \cos \dfrac{\theta}{2}$

$qr_x = \omega_x \times \sin \dfrac{\theta}{2}$

$qr_y = \omega_y \times \sin \dfrac{\theta}{2}$

$qr_z = \omega_z \times \sin \dfrac{\theta}{2}$

Given initial three-dimensional angular positions (orientation) $qs$, it can be said that moving at angular velocities $\omega_x$, $\omega_y$ and $\omega_z$ for a period of time $t$ sees us ending up at final three-dimensional angular positions (orientation) $qs'$:

$qs' = qs \times qr$

Therefore, as straight line displacement $\delta s$ can be accumulated to track the position along a straight line $s$, so can angular displacements about three axes (or, rather, their single equivalent angular displacement about the Euler axis $qr$) be accumulated to track angular position in three-dimensional space (orientation) $qs$.