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I was wondering if there is any good way to estimate the difference between two joint states of a robotic arm.

Currently I am using something like this: $\sqrt{\sum_i^N{(JStateA[i]-JStateB[i])^2}}$. Where $JStateA$ and $JStateB$ are the two joint states.

However, this formula gives the same weight to all joints. Ideally, I would like the formula to reflect that moving the first joint (that moves the whole arm) is a much more significant change than moving the last joint (which only moves the hand).

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I'm guessing you're wanting to change the weighting to modify some joint optimization code? But if you want to change the weighting then you need to decide how those weights change.

The best advice I think I could provide, given the wording of your question, would be to consider the Jacobian for your robot arm. The naive approach would be to consider just distance between the joint and end effector, but that doesn't consider the configuration of your robot arm.

The Jacobian puts all of your partial derivatives together in one place for you, and that gives you how much your end effector changes given a change in a particular joint value - $\partial f/\partial q$.

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