If we let:
a(t) = a0 +w * t
where:
The quaternion representation of a rotation angle (orientation) is:
q = cos(angle/2) + i sin(angle/2)*axis_{x} + j sin(angle/2)*axis_{y} + k sin(angle/2)*axis_{z}
where:
so if the quaternion is a function of time we substitute angle for a0 +w * t to give:
q = cos(( a0 +w * t)/2) + i sin(( a0 +w * t)/2)*axis_{x} + j sin(( a0 +w * t)/2)*axis_{y} + k sin(( a0 +w * t)/2)*axis_{z}
So this gives the orientation as a function of time, so to get angular velocity we need to differentiate this with respect to time, as discussed here, so we differentiate q to get qw which is a quaternion representing angular velocity.
So if we differentiate each term we get,
real terms > d(cos(( a0 +w * t)/2))/dt = d(cos(( a0 +w * t)/2))/d( a0 +w * t)/2) * d( a0 +w * t)/2)/dt = sin( a0 +w * t)/2)*w
imaginary terms > d(sin(( a0 +w * t)/2))/dt = d(sin((a0 +w * t)/2))/d(( a0 +w * t)/2) * d(( a0 +w * t)/2)/dt = cos((a0 +w * t)/2)*w
So if we put the quaternion back together we get:
qw = dq/dt = w * (sin(( a0 +w * t)/2) + i cos(( a0 +w * t)/2)*axis_{x} + j cos(( a0 +w * t)/2)*axis_{y} + k cos(( a0 +w * t)/2)*axis_{z})
So what is the physical interpretation of this? Unless I have made an error (which is very likely) an instantaneous rotation velocity can be represented by a quaternion but its elements would be continuously varying with time in a way that is difficult to interpret?
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