# How to derive the Jacobian for a floating base linkage?

I am trying to determine the Jacobian for a planar bipedal robot. I'm modeling it as a floating base system with 5 links (floating base and 2 two-link legs), which seems to be the common approach. I know that the dimension of the Jacobian must be 5 by 3 since I have 5 links and 3 degrees of freedom in the plane. However, I'm not sure how to proceed because I've only dealt with linkages that are always connected to the base. Any help would be appreciated. Thank you.

Actually since a Jacobian maps joint velocities to workspace velocities (linear and angular) of a point of interest (for example, the robot's center of gravity), in this case, your Jacobian will have the dimension of $$3 \times 4$$, where 3 is the number of your workspace degrees-of-freedom (2 translation axes + 1 rotation axis) and 4 is the number of joints.