Physics - Kinematics - Angular Velocity ()

Representing angular velocity using quaternions

If we let:

a(t) = a0 +w * t


The quaternion representation of a rotation angle (orientation) is:

q = cos(angle/2) + i sin(angle/2)*axisx + j sin(angle/2)*axisy + k sin(angle/2)*axisz


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)*axisx + j sin(( a0 +w * t)/2)*axisy + k sin(( a0 +w * t)/2)*axisz

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)*axisx + j cos(( a0 +w * t)/2)*axisy + k cos(( a0 +w * t)/2)*axisz)

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?

metadata block
see also:
Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.


cover Engineering Mechanics - Includes Statics book and dynamics book below..

cover Engineering Mechanics Vol 2: Dynamics - Gives theory for rigid dynamics, aims to allow prediction of effects of force and motion. Includes rotating frame of reference. Lots of colour diagrams, I guess its college / University level.

Commercial Software Shop

Where I can, I have put links to Amazon for commercial software, not directly related to the software project, but related to the subject being discussed, click on the appropriate country flag to get more details of the software or to buy it from them.

cover Mathmatica

This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2023 Martin John Baker - All rights reserved - privacy policy.