Question: Symbolic calculation of matrix rank

Hello,

I would like to symbolically determine the rank of a jacobian matrix. In the help, I have seen that the Rank function of the LinearAlgebra can be used for this purpose. However, when I use this function, the function doesn't allow to find the different singularities that can occur on my jacobian matrix.

Here a exemple of a jacobian matrix that I obtain on a slidercrank mechanism:

The rank of this jaobian (Phi) gives 2 whatever the values of theta(t) and beta(t). However, if the values of  theta(t) and beta(t) are :theta(t)=Pi/2,beta(t)=0. The rank shouldn't be 2 but 1.

Is a way to obtain the symbolic calculation of the rank of a jacobian matrix which can distinguish different cases following the values of the parameters ? In others words, my dream will be to have a Rank function (or another algorithm) which can gives :
the rank is 2 if theta(t) different of Pi/2 [Pi] and beta(t)=0 [Pi] 
and otherwise 1 if ...
and perhaps 0 if ...

Thanks a lot for your help.

I let a piece of code with an example of calculation of the rank

RankMatrix.mw