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From: "Bruno Luong" <b.luong@fogale.findmycountry>
Newsgroups: comp.soft-sys.matlab
Subject: Re: Find instant axis of rotation
Date: Wed, 28 Oct 2009 07:18:00 +0000 (UTC)
Organization: FOGALE nanotech
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"Henning Olsson" <asd@prutcom.com> wrote in message <hc7mhj$69k$1@fred.mathworks.com>...
> I need to estimate/calculate the location of a rotation axis for a number of points in space. I have the position of the all of points (x,y,z) at all of the different times. The relative location of all the points remain the same as they are all part of a rigid body.
> 
> I started out by creating a rotation matrix around the origin (by using 3 points in step n and step n+1 and then solving for the rotation matrix). I then used an Axis-Angle approach to calculate the axis around which they rotate.
> 
> The problem is that I am pretty sure that for the actual motion the axis doesn't go through the origin but is translated somehow in space. But I'm not sure how to find this translation. Anyone have any ideas?

1. Rotation axis (direction only) R is perpendicular to the velocity field. So estimate R is easy, for example as following:
       Minimize Sum | Projection R on V_i |^2 such that |R|=1
      | Projection R on V |^2 = <R,V>^2 / | V |^2

2. Project the velocity to the plane perpendicular to R, V projected is tangential to the circle centered about the axis center. The magnitude is proportional to the radius. Write this as equation than solve it for he axis center.

Bruno