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The quaternion algebra shows there as a way of disentangling two Alamouti coded signals transmitted by a pair of antennas. The advantages come from the fact that even if the signal from one antenna is lost for a particular receiver (due to sitting in a node for that particular radio wave), then the signal from the other antenna saves the day.

Oct 19, 2010 · Here is the intuitive interpretation of this. Given a particular rotation axis $\omega$, if you restrict the 4D quaternion space to the 2D plane containing $(1,0,0,0)$ and $(0,\omega_x,\omega_y,\omega_z)$, the unit quaternions representing all possible rotations about the axis $\vec \omega$ form the unit circle in that plane.

May 27, 2020 · Adding two unit quaternions generally does not yield a unit quaternion, so the answer is technically no as written, but the answer is yes if you say "rotating two separate planes by the same angle and rescales." Of course adding two quaternions gives a quaternion, so algebraically this is clear.

where log/exp are the quaternion logarithm/exponential. If you can convert between rotation vectors (where the direction is the rotation axis and the magnitude is the angle) and quaternions you can use that as the exponential, and the logarithm is the reverse operation (both modulo a factor 2 that cancels out).

If I combine 2 rotation quaternions by multiplying them, lets say one represents some rotation around x axis and other represents some rotation around some arbitrary axis. The order of rotation ma...

In the quaternion case, reduced means that instead of taking this as the norm, you take its square root. Since the quaternions are 4-dimensional over $\Bbb R$, the reduced norm defines a quadratic form, which is what one would expect from an euclidean norm. $\endgroup$

May 24, 2016 · First: note we are dealing only with the unit quaternions as a representation of attitude. The full quaternions don't really have a role here. I should also note up front that the quaternion itself has a rate ($\dot{q}$), but like the Euler angle rates the quaternion rate is not the actual angular velocity, which is a 3-vector. They are ...

May 4, 2018 · But if I apply this to the Quaternion I have it looks as if it is just subtracting each ijk from each other, or maybe I'm just not understanding this correctly. I'd like to better understand what is going on here and how to get the conjugate of a Quaternion properly. I'm happy to follow any web links that better explain the method of doing this.

Aug 5, 2015 · Every quaternion multiplication does a rotation on two different complex planes. When you multiply by a quaternion, the vector part is the axis of 3D rotation. The part you want for 3D rotation. But you ALSO do a rotation in the complex plane consisting of …