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Subject: Re: How to solve TA = BT
Date: Mon, 19 Jan 2009 18:26:01 +0000 (UTC)
Organization: Xoran Technologies
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"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in message <gl2f6r$91v$1@fred.mathworks.com>...
> "Matt " <mjacobson.removethis@xorantech.com> wrote in message <gl29kl$qn5$1@fred.mathworks.com>...
> > .......
> > In addition to what Roger said, T has 6 degrees of freedom, so if n>6, it is likely that the above system of equations will be overdetermined. So, I'm assuming that what you want is some sort of least squares solution
> > 
> > (1)  min.  sum_i || T*Ai-Bi*T ||^2   
> > 
> > If you parameterized T in terms of angles and translations, you could solve this using fmincon(), althrough you would want a good initial guess of T as insurance against local minima.
> > .......
> 
>   Even with n = 1, it is overdetermined in that sense, Matt.  The equality
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Well, not always. Suppose n=1 and A=B=I. Then the equations reduce simply to
T=T and you have as undetermined a problem as you can get.



>   With reference to the least squares approach, I think you meant 'fminunc'.  There would be no constraints in that case with the six parameters - any Euler angles and any displacements are valid cases to be considered in the minimization process.

I suppose. But there's also the possibility that you might wish to constrain the angles to [0,2*pi] to avoid redundant solutions...