The mechanism shown on the picture has prismatic, revolute and spherical joints in order and I derived the screw based Jacobian in local frame $B_i$, for the isolated leg. The isolated leg coordinate frame is located at $B$ and it is a copy of frame $O$.The leg is assumed to have a center of End-effector at $A_i$. Now, I suppose to transform the Jacobian to O' and O. and I am confused. Can anyone help me, please? Here is the form HMT used to drive Jacobian $ H^B_1H^1_A$, which is from $B_i$ to $A_i$ and the $\\ Ad_g=\begin{bmatrix}R&[P]_xR\\0&R\end{bmatrix}$ is the matrix I used to premultiply the Jacobian. But what I make me confused is the path to transforming the given Jacobian to frame $O$ and $O'$.

AS it can be seen on the image there are four interest coordinate frames. These are <span class=$O$,$B$,$A$ and $O'$ ">


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