Unsure how to interpret Minerr result...

My question is, how do I interpret the resultant Matrices/Vector in the result set? In other words, how do I determine u1, u2, u3, SX, SY and SZ?

Simple answer - as you have set it up you can't. You provide a guess value for the matrix S which is a diagonal matrix, but as you can see does Mathcad not respect your will that S[0,1 should be zero, etc. YOu will have to state that as additional constraints in the solve block. As those constraints aren't hard constraints, you may even write 10^5*S[0,1=0, thus weighting that constraintheavily and chances are the value will come out close to zero at the end.

Similar your matric P. For Mathcad you simply provide 9 arbitrary guess values and Mathcad feels free to play with all of them. So again you have to constrain some of them to zero and to keep the relationship between -sin(u1) and cos(u1) you may write U1 and sqrt(1-U1^2) instead. Same with U", U3 and sqrt(1-U2^2-U3^2).

That way you will have 30 equations / constraints and 21 variables to solve for.

Alternative would be to just use Minerr(SX,SY,SZ,BX,BY,BZ,u1,u2,u3) , which would mean 20 equations and 9 variables to solve for. This time you would have to write b=(BX,BY,BZ) (boolean equal sign) in the solve block instead of b:=(BX,BY,BZ) above. Same for the other matrices.

A quick try yielded the attached (I'm not able to save in 2001 format, so I attach a pdf) which is less than disappointing. ERR is extremely high and setting CTOL to something lower than the high value set in the sheet (0.1) does not change anything. So either there is really no better fit or I have done something wrong.