# Elegant way to mathematically relate joint positions (without a root/base joint) for easy re-rooting?

I'm pretty new to robotics, but have learned about the tranformation matrices and how to relate relationships between joints using transformation matrices.

I just wonder if there is a system that represents these relationships without having to choose a root node?

The reason I'm asking is for re-rooting of an arm.

If we think of a simple robotic arm in 2-D, we usually have a base or root joint, then a few links and an end-effector.

We can then compute each joint position in the global (root) reference frame using the transformation matrices for each link.

But what if we want to suddenly switch and fix the end-effector as the new root/base and move the old root (as the new end effector)?

Normally, we need to re-create the transformation matrices from the new root out to the new end effector.

Is there a more elegant way to do this?

If $$T^{ee}_{root}$$ is the homogeneous transformation that relates coordinates expressed in the end-effector frame into coordinates expressed in the root frame, then it holds that $$T^{root}_{ee} = \left( T^{ee}_{root} \right)^{-1}$$.