How to find inertia parameter of payload automatically

The parallel axis theorem gives:

$$ I = I_{textrm{cm}} + md^2 \ $$

where $I_{textrm{cm}}$ is the moment of inertia about the object's center of mass, $m$ is the mass of the object, and $d$ is the distance from the center of mass of the object to the axis of rotation.

You can also look at load acceleration and torque, where:

$$ tau = Ialpha \ $$

Or, torque is moment of inertia times angular acceleration. Then, you could estimate

$$ I = frac{tau}{alpha} \ $$

Apply a torque, measure the angular acceleration, and divide the two to get load inertia. You could assume the center of mass is located at the tip of the end effector, which isn't true, but as long as the load is fixed it doesn't really matter because the effective moment of inertia is what the arm's actually controlling. Consider it like:

$$ I_{textrm{apparent}} = I_{textrm{effective}} + md^2\ $$

where $d$ is the distance from the joint to the tip of the end effector, and $I_{textrm{effective}}$ is

$$ I_{textrm{effective}} = I_{textrm{cm}} + md_{textrm{offset}}^2 \ $$

where $d_{textrm{offset}}$ is the difference between the object's center of mass and the object's connection to the end effector. Again, the arm actually moves the effective moment of inertia of the load.

You could also determine the mass by the holding torque required to keep the arm stationary, where

$$ tau = mgdsin{theta}\ $$