I am currently trying to parametrize the low-level gains of a robotic arm. This arm uses a classical PID for each joint.
I am trying to use a method based on computation rather than a trial-and-error/tweaking approach. The method I use considers each joint independently and assumes the system driven by the PID is linear. Hence I infer a transfer function, a characteristic polynomial, poles and this gives me gains $K_p$, $K_i$, and $K_d$ for each joint.
Now, computed as I did, these gains depend on the natural angular-frequency. For example: $$ K_p = 3 a w^2 $$ where $a$ is the inertia and $w$ is the natural angular-frequency.
Hence my question: how shall I compute $w$, the natural angular-frequency for my system? Is this an enormous computation involving the geometry and other complex characteristics of the robot, and that only a computer can do or are there simple assumptions to be made which can already give a rough result for $w$?
I guess this is a complex computation and this is one of the reasons why PID gains are most often found by trial-and-error rather than from computation. Though I am looking for some more details on the subject to help me understand what is possible and what is not.