# solving inverse kinematic using DH to keep it rotating over a center of motion

I am working on a 7 DOF robot. I want to solve the Inverse Kinematic(IK) of the robot using the inverse transformation matrix which I found them using DH. I mean: $$\begin{equation*} T = \begin{pmatrix} R & p \\ 0 & 1 \\ \end{pmatrix}\\ T ^{-1}= \begin{pmatrix} R^{T} & -R^{T}p \\ 0 & 1 \\ \end{pmatrix} \end{equation*}$$ I am a little bit confused about that and I have these questions:

1)Does this method gives a unique solution, eventhough there is 7 joints?

2)How to avoid singularities by this method?

3)If I want to have a center of motion on my robot(lets say on link 2, that the robot always stays on that point and rotate over it, like a surgerical robot that only should go through the body over a single small hole), How can I consider that in my IK solution?
If I put my first three frames of DH on that point, does that make the IK solution consider that constraint automatically?