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Some questions about this, my friends and I argued with this problem.

Are operational space and joint space dependent on each other?

I know that $x_e$ (end effector's pos.) and $q$ (joint var.) can be expressed by an equation with non-linear function $k$:

$x_e = k(q)$

But I don't think that it tells us operational space and joint space are dependent.

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What an interesting question. It's interesting because operational space parameters are the same if a robot is in configuration 1, or configuration 2, or even if it is not present at all. Similarly, the robot can choose any joint space configuration it desires, independent of any operational space parameters. So it would seem these two spaces are completely independent.

It is the act of having the robot perform tasks that relates and constrains the two spaces. Once you task a robot to "move to point A (in operational space)," you have just built an implicit constraint on the robot's joint space such that it must use the $\vec x = K (\vec q)$ kinematics to move to point A. So any function of operational space tasks $f(\vec x$) gets mapped via inverse kinematics $K^{-1}$ to constrain joint space $\vec q$.

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