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In article <3784@news.cerf.net>, heimbold@nic.cerf.net (John Heimbold) writes:
|> In article <6611@dciem.dciem.dnd.ca> sjm@dciem.dciem.dnd.ca (Stewart Moorehead) writes:
|> >
|> > I am recording the 6D position (x,y,z,roll,pitch,yaw) of an object
|> > using two independant devices. Since these devices have different
|> > reference frames a transformation matrix, call it X, must be
|> > calculated. To calculate X, the equation AX=XB must be solved, with A
|> > and B being known 4x4 matrices. I have found two methods which claim to
|> > solve for X. They were proposed by Roger Tsai and Reimar Lenz (June
|> > 1989 IEEE Transactions on Robotics and Automation) and by C. C. Wang
|> > (April 1992). Both of these methods work if I pre-define X and create
|> > some 'cooked' data around it, however when I use actual data that was
|> > recorded using teh devices I get an incorrect answer. The devices being
|> > used are any two of : Northern Digital's Optotrak, Logitech's 3D mouse,
|> > and Polhemus Fastrak. I feel that the inherent error in these devices
|> > may be too great for the methods being used.
|> > I would appreciate it if anyone has information regarding these methods
|> > or the devices being used. If anyone has solved a similar problem
|> > please respond.
|> > You can send mail to sjm.dretor.dciem.dnd.ca
|> 
|> The problem as you stated it may or may not be well defined.
|> First of all, as stated, there may be many solutions for X
|> analogous to finding a set of eigenfunctions for X.  To see this,
|> consider A=B.  In this case X=0 is a solution as well as X equal
|> to the identity matrix.  This is one example of multiple solutions
|> to the equation.
|> 
|> Next, consider the problem definition.  Since you want to estimate
|> X, try a statistical estimation approach.  Formulate the problem
|> by doubling the number of measurements (assumming that they are
|> noisey) and then figure how to use this information to estimate
|> X.  A similar category of problem is in photogrammetry where one
|> estimates the location and orientation of a camera given the 
|> photograph and some measured points in the "true" scene.

Consider the following:
	A*X = X*B
is equivalent to 
	(A (x) I  -  I (x) B) * vec(X) = 0
where (x) is the Kronecker product and vec() is an operation which
stacks the columns of X into a vector.  Thus, the equation is a linear
system of 16 equations in 16 unknowns.  If A (x) I - I (x) B is full
rank then X=0 is the only solution.  Otherwise, the solution set is the
set of matrices X for which vec(X) spans the kernel of A (x) I - I (x) B.

Check out the following paper:

    author=     {P. Lancaster},
    title=      {Explicit Solutions of Linear Matrix Equations},
    journal=    {SIAM Review},
    volume=     {12},
    number=     {4},
    pages=      {544-566},
    month=      {October},
    year=       {1970}

Matt Wette
-- 
mwette@csi.jpl.nasa.gov