I have been given a Jacobian for a 3 revolute joint robot.
I need to obtain the origins of each of the joint. Origins of the joints can be obtained once I get $T^0_1$, $T^0_2$, $T^0_3$ and $T^0_4$. 4th frame is my end effector.
Is there any way to calculate the transformation matrices from the Jacobian? I have gone through literature extensively and I can get the Jacobian from the transformation matrices using the below relations
The Jacobian is
$$J = \begin{bmatrix} J_v\\ J_w \end{bmatrix}$$
where for revolute joints,
$$ J_{v_i} = z_{i-1} \times (o_n - o_{i-1}) \\ J_{w_i} = z_{i-1}$$
Is the reverse possible i.e, obtaining the transformation matrices when I have the Jacobian?
Is there any way to obtain just the origins of the frames $ o_1, o_2, o_3$ ($o_0 = \begin{bmatrix} 0 &0 &0 \end{bmatrix}^T$ ) from the given Jacobian? The reverse cross product method does not give me the complete co-ordinates of the origin.